– Typeset by GMNI & FoilTEX –

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

a: Dept. of Applied Mathematics Civil Engineering School University of A Coru˜ na, Spain

INITE

OLUME

X. Nogueiraa, L. Ram´ıreza, S. Clainb, R. Loub` erec, L. Cueto-Felguerosod, I. Colominasa

Universidade do Minho Portugal

LOCAL

COMMITEES:

b:Centre of Mathematics University of Minho Guimar˜ aes, Portugal Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

An SPH method with MOOD limiting for compressible flows

c: Institut de Math´ ematiques Universit´ e de Toulouse Toulouse, France d: Dept. of Civil Engineering UPM Madrid, Spain

e-mail: [email protected] web page: http://caminos.udc.es/gmni

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

•

Introduction

•

The SPH-ALE method

•

MLS approximations

•

The Multi-dimensional Optimal Order Detection (MOOD) Method

•

Application to Euler equations

•

Conclusions

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Outline

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

•

Introduction

•

The SPH-ALE method

•

MLS approximations

•

The Multi-dimensional Optimal Order Detection (MOOD) Method

•

Application to Euler equations

•

Conclusions

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Outline

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

A Coru˜ na...OK. But... where is it?

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

A Coru˜ na...OK. But... where is it?

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

A Coru˜ na...OK. But... where is it?

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

A Coru˜ na...OK. But... where is it?

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

J.P. Vila, On Particle Weighted Methods and Smooth Particle Hydrodynamics, Mathematical Models and Methods in Applied Sciences, 9 (2), 161-209, 1999.

INITE

OLUME

I Origin of this research: • SPH formulations relying on the artificial viscosity approach present significant loss of accuracy in flows involving shocks or contact discontinuities. • SPH Riemann-based methods couples a Riemann solver with the SPH framework. • Development of an accurate, stable and low-dissipative SPH Riemann-based method. • Formulation based in the SPH-ALE scheme developed by Vila1.

1

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Introduction

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• • • • • • Discretization using a set of arbitrarily distributed particles. Integral function approximation: kernel approximation. Particle approximation of field functions. Summation to replace integration. No connectivity is defined between particles. Solve by using explicit integration algorithms.

Grid-based Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: SPH basics

Particle-based

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I We start from the equality:

x − y ) by a smoothing function I We replace δ(x W (x − y, h) (kernel) x)i = hu(x

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

x) = u(x

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: Integral representation

Z y ∈Ω

y ∈Ω

x − y )dy u(yy )δ(x

Z

x − y , h)dy u(yy )W (x

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

I Compact condition W (x x − y , h) = 0 ∀|x − y| > κh

INITE

OLUME

I Unity condition: Z

I Delta function property lim W (x x − y , h) = δ(x − y) Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

y ∈Ω

h→0

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

Introduction: Properties of kernel functions

x − y , h)dy = 1 W (x

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I The problem domain is discretized using a finite number of particles. I We replace the infinitesimal volume dy in previous integrations at the location of particle j by the finite volume of the particle (Vj ). I Then, each particle occupies a volume Vj and carries a mass (mj = ρj Vj ). x)i = hu(x

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

LOCAL

y ∈Ω

Universidade do Minho Portugal

Z

x − y , h)dy ∼ u(yy )W (x = Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

Introduction: Particle approximation

= N X

j=1

N X

x − yj , h)Vj u(yy j )W (x

j=1

mj x − yj , h) u(yy j )W (x ρj

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

N X mj x)i = − x − yj , h) h∇ · u(x u(yy j )∇W (x ρj j=1

hu(x x)i =

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: Particle approximation

N X

j=1

mj x − yj , h) u(yy j )W (x ρj

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Many functions used as kernels: splines, gaussians I An example, the cubic spline:   s ≤1 1 − 32 s2 + 34 s3   α  1 3 Wj (x x) = W (x x − y j , h) = ν (2 − s) 12

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: Particle approximation

s= kx−y j k h



h = k max ( x − y j )

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Another example: Exponential Kernel.

s = |y − x∗| , dm = 2 max (|yj − x∗|) , c =

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

W (y, x∗, κ) =

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: Particle approximation

e −( sc ) 2

−e 2 −( dm ) c 1−e

−( dm c ) 2

dm 2κ

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Another example: Exponential Kernel.

s = |y − x∗| , dm = 2 max (|yj − x∗|) , c =

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

W (y, x∗, κ) =

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: Particle approximation

e −( sc ) 2

−e 2 −( dm ) c 1−e

−( dm c )

2

dm κ 2κ

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

CUBIC SPLINE

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: Particle approximation

EXPONENTIAL KERNEL

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I We consider the 1D Euler equations  ∂ρ ∂(ρu)   + =0    ∂t ∂x     ∂(ρu) ∂(ρu2 + p) + = 0  ∂t ∂x       ∂(ρE) ∂(ρEu + pu)   + =0 ∂t ∂x Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: SPH method

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• Following Raviart1, separating the convective terms from pressure terms in the flux term of the Euler equations, and then applying the particle approximation, we get, for each particle i:                  

   N  X  d pi pj   xi − xj , h) = 0 (ViρiEi) + Vi Vj ρj ui + ρiuj ∇W (x   dt ρ ρ  i j  j=1         dxi = ui dt Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

d Vj (Viρiui) + Vi dt j=1

d (Viρi) = 0 dt

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

Introduction: SPH method

N X

 pi pj xi − xj , h) = 0 ρj + ρi ∇W (x ρi ρj



P.A. Raviart, Particle approximation of first order systems, Journal of Computational Mathematics, 4 (1), 50-61, 1986. 1

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• Taking in account the first equation, we arrive at

 ! N X  du pi pj  i  xi − xj , h) = 0 + + 2 ∇W (x m  j 2   dt ρi ρj  j=1     !  N dEi X pi pj + mj xi − xj , h) = 0  2 uj + ρ2 ui ∇W (x  dt ρ  i j  j=1         dxi = ui dt Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: SPH method

mj = Vj ρj

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• In standard SPH codes, a pseudo-viscous pressure dissipation is added for stabilization  ! N X  du pi pj  i  x + mj  2 + ρ2 + Πij ∇W (x i − x j , h) = 0   dt ρ  i j j=1     !  N pj dEi X pi xi − xj , h) = 0 u + ui + Πij ∇W (x + m  j j 2 2  dt ρi ρj   j=1         dxi = ui dt Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Introduction: SPH method

mj = Vj ρj

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

•

Introduction

•

The SPH-ALE method

•

MLS approximations

•

The Multi-dimensional Optimal Order Detection (MOOD) Method

•

Application to Euler equations

•

Conclusions

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Outline

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Instead of using artificial viscosity, it is possible to use Riemann solvers. I Here we follow the method developed by Vila, connected with the Arbitrary Lagrangian-Eulerian (ALE) approximation. I The 2D Euler equations, cast in a general system of conservation laws in a Lagrangian frame, write U ∂U F − v f rame ⊗ U ) = 0, + ∇ · (F ∂t Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

in Ω,

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• Instead separating the convective terms from pressure terms in the flux term of the Euler equations, we can write2, for each particle i:

N X d xi − xj , h) = 0 (ViΦi) + V i Vj (F (xj , t, Φj ) + F (xi, t, Φi) + Πij ) ∇W (x dt j=1

2

J.P. Vila, On particle weighted methods and SPH, Mathematical Models and Methods in Applied Sciences, 9 (2), 161-209, 1999.

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

NOW-HOW on

B Gij is the numerical flux computed at point r ij = (rr i + r j )/2. B H i = F (U U i) − v i ⊗ U i is the Lagrangian flux tensor computed at the state of the i−th particle (flux difference formulation of the SPH scheme3). This term can also be seen as a correction for the computation of the geometrical weights of the SPH-ALE approximation 4.

3

D. Avesani, M. Dumbser, A. Bellin, A new class of Moving-Least-Squares WENO-SPH schemes, Journal of Computational Physics,270, 278-299, 2014.

OLUME

B Instead of using Πij Vila proposes the use of Riemann solvers.

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

ni X dViU i Gij − H i) · ∇Wij = − ViVj 2 (G dt j=1

i dVi X = ViVj 2 (vv ij − v i) ∇Wij dt j=1

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

INITE

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

n

4

M. Schaller, R. G. Bower, T. Theuns, On the use of particle based methods for cosmological hydrodynamical simulations, 8th international SPHERIC workshop, Trondheim, Norway, June 4-6, 2013. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

B Instead of using Πij Vila proposes the use of Riemann solvers.

B Written in this form, the method does not keep Viρi = mi, but global conservation of mass, momentum and energy is kept. Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

ni X dViU i Gij − H i) · ∇Wij = − ViVj 2 (G dt j=1

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

dVi = dt

ni X

ViVj 2 (vv ij − v i) ∇Wij

j=1

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I The evolution of the particle position is given by a weighted average interpolation

I We use Wij = 0.5(Wij (hi) + Wij (hj )) Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

ni X

Vj v j Wij

drr i j=1 = ni X dt Vj Wij

j=1

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

I where qij is defined as qij = krr j − r ik /hij , αd is a normalization constant so the unity of the partition of the unity is verified, αD = 1/hij in 1D and αD = 15/7πh2ij in 2D, and hij is the smoothing length.

INITE

OLUME

I We use the cubic spline to compute Wij and ∇Wij  3 3 3 2   1 − qij + qij   2 4   1 Wij = αd 3 (2 − q )  ij   4    0

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

qij ≤1

1 < qij ≤2

qij >2

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

1

J.J. Monaghan and H. Pongracic, Artificial viscosity for particle methods, Applied Numerical Mathematics 1 :187-194,1985.

INITE

OLUME

I In order to increase the accuracy of the SPH-ALE method, we improve the reconstruction of the variables for the Riemann problem. I High-order reconstructions are performed using MLS approximations I However, the global order of the scheme is 2, independently of the order of the reconstruction, due to the discretization of the divergence of the convective terms in the Euler equation1. Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

•

Introduction

•

The SPH-ALE method

•

MLS approximations

•

The Multi-dimensional Optimal Order Detection (MOOD) Method

•

Application to Euler equations

•

Conclusions

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Outline

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

• W (zz − yy, h) |z =x : kernel (smoothing function) with compact support (Ωx ) centered in z = x. • h: smoothing length.

INITE

OLUME

x) at a point x by using a I Reconstruction of u(x weighted LS approximation in the vicinity of x: x) ≈ u u(x ˆ(x x) =

x): base of functions with dimension m. • pT (x • α (zz ) |z =x : Parameters that minimize the error functional: J(α α(zz ) |z =x ) =

Universidade do Minho Portugal

LOCAL

COMMITEES:

y ∈Ωx

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

i=1

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (I)

m X

Z

α(z) |z = x pi(x x)αi(zz ) |z =x = pT (x x)α(z)

 2 T W (zz − y , h) |z =x u(yy ) − p (x x)α α(zz ) |z =x dΩx

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Minimization of J leads to: Z

x)α p(yy )W (zz − y , h) u(yy )dΩx = M (x α(zz ) y ∈Ωx z =x z =x

x) is the moment matrix defined as: I M (x x) = M (x p(yy )W (zz − y , h) pT (yy ) y ∈Ωx z =x Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (II)

Z

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I In practice, Ω is a set of scattered points. Previous integrals are evaluated using points in Ωx as quadrature points:

• uΩx contains nodal values of the function ux to be approximated, at nx nodes in Ωx COMMITEES: LOCAL

uΩx = u(x x1) u(x x2) · · · u xnx Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

x)u uΩx x)P α (zz ) = M −1(x P Ωx W (x z =x

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

Moving Least Squares (III)



T

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

I Finally, MLS approximation is written by:

INITE

OLUME

I Discrete expression of the moment matrix is a m × m x) = P Ωx W (x matrix equals to M (x x)P P TΩx x)(dimension nx × nx ) are • P Ωx (dimension m × nx ), and W (x obtained by



W (x x) = diag {Wi (x x − xi)} i = 1, . . . , nx

x)u uΩx = N T (x x)u uΩx = x) = pT (x x)M M −1(x x)P P Ωx W (x u b(x Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

x2) · · · p xnx x1) p (x P Ωx = p (x

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (IV)

nx X

x)uj Nj (x

j=1

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

(1)

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

I Nj can be considered as “shape functions”. I Nj depends on the number of neighbors, the kernel and the base (ppT ). I Nj is a function of the grid.

INITE

OLUME

I Interpolation can be written as:

with

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

u ˆ(x x) =

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (V)

nx X

x)uj Nj (x

j=1

x) x) = pT (x x)M M −1(x x)P N T (x P Ωx W (x

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I A practical note about the polynomial basis

• We define locally and scale the monomials of the basis • Better conditioning of the momentum matrix x) are evaluated at a point xI , • If MLS shape functions N (x the basis is evaluated at x−hxI • Then we can write: M −1(x xI )P P Ωx W (x xI ) = pT (00)C C (x xI ) N T (x xI ) = pT (00)M Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

with

p(x x) = 1

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (VI)

x y xy 2

x

y
2 T

I

xI ) P Ωx W (x xI ) = M −1(x xI )P C (x I

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Computation of derivatives • First derivatives C (x x) ∂C x) ∂ppT(x x) ∂N N T(x T x) x) + p (x = C (x ∂x ∂x ∂x

• Second derivatives

x) ∂C C (x x) ∂ 2N T(x x) ∂ 2pT(x x) ∂ppT(x ∂ 2C (x x) x) + 2 x) = C (x + p(x 2 2 ∂x ∂x ∂x ∂x ∂x2 C (x C (x x) ∂ 2pT(x x) ∂ppT(x x) ∂C x) ∂C ∂ 2N T(x x) x) ∂ppT(x x) + = C (x + ∂x∂y ∂x∂y ∂x ∂y ∂y ∂x 2 x) ∂ C (x T x) + p (x ∂x∂y Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (VII)

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• where ∂C∂x(x) is given by
x) C (x ∂W W (x ∂C x) −1 T W (x x)C x)W x) x) C (x = C (x I − p (x ∂x ∂x

x) • and the second derivatives of C (x ∂ 2C (x x) ∂x2

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

=

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (VII)

 ∂C C (x x) −1 ∂W  T C (x W (x x) I − p (x x)C x) ∂x ∂x  ∂ 2 W (x x)  −1 T C (x C (x W (x I − p (x x)C x) +C x)W x) ∂x2  ∂W (x x) −1 ∂W (x x)  T −1 C (x C (x W (x W (x x) I − p (x x)C x) −C x)W x) ∂x ∂x ∂W (x x) T ∂C C (x x) C (x W −1 (x p (x x) −C x)W x) ∂x ∂x

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

x) ∂ 2C (x ∂x∂y

• The diffuse derivatives are obtained by neglecting all derivatives of C (x x)  ∂ 2 W (x x)  T W (x C (x C (x x) x)W +C x) x)C I − p (x ∂x∂y  ∂W (x x) −1 ∂W (x x)  −1 T C (x W (x C (x −C x)W x) x) x)C x) W (x I − p (x ∂y ∂x

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

=

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (VII)

 ∂W (x x)  x) −1 ∂C C (x T C (x x) x)C x) W (x I − p (x ∂y ∂x

W C (x x)W −C

−1

−1

(x x) ∂W (x x) T x) ∂C C (x x) p (x ∂x ∂y

x) ∂ 2pT(x ∂ 2N T(x x) x) ≈ C (x 2 2 ∂x ∂x x) ∂ 2N T(x x) ∂ 2pT(x x) ≈ C (x ∂x∂y ∂x∂y

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• However, as we already have computed the first derivatives x), it is possible to use a semi-diffuse approach of C (x without extra effort: x) x) ∂C C (x x) ∂ 2N T(x ∂ 2pT(x x) ∂ppT(x x) + 2 ≈ C (x 2 2 ∂x ∂x ∂x ∂x C (x C (x x) ∂ppT(x x) ∂C ∂ 2N T(x x) ∂ 2pT(x x) x) ∂ppT(x x) ∂C x) + ≈ C (x + ∂x∂y ∂x∂y ∂x ∂y ∂y ∂x

• It has been proved that use of diffuse or semi-diffuse derivatives does not decrease the order of accuracy. It should be noted that the accuracy is affected Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Moving Least Squares (VII)

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Note that due to the local scaling

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

pT(00) = (1

T

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The MLS method (I)

∂pp (00) = ∂x



∂ppT(00) = ∂y

 0

0

0 0

1 h 0

0 1 h

0

0

0

0

0

0

0)



0



0

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

I The approximation basically depends on a kernel and a basis function.

INITE

OLUME

I MLS is an approximation method very used by the meshless community. x) at a point x I MLS performs a reconstruction of u(x by using a weighted LS approximation in the vicinity of x. I The approximation is written in terms of MLS shape functions. Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

The MLS method in a nutshell

x) = u ˆ(x nx X

Nj (x x)uj

j=1

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

•

Introduction

•

The SPH-ALE method

•

MLS approximations

•

The Multi-dimensional Optimal Order Detection (MOOD) Method

•

Application to Euler equations

•

Conclusions

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Outline

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

NOW-HOW on

OLUME

Ui

• The fundamental idea behind the MOOD paradigm is to determine, a posteriori, the optimal order of the polynomial reconstruction for each particle that provides the best compromise between accuracy and stability. • The MOOD method is based on two main ingredients: a Particle Polynomial Degree (PPD), and a chain of detectors to assess the validity of the candidate solution.

ri n

n

SPH solver

artif.visc. slope...limit.

STABILIZATION

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

INITE

A priori SPH code

Ui

ri n+1 ri n

n+1 n

Ui

SPH solver

di = d i − 1 BAD

cubic scheme base scheme

DECREMENTING

CANDIDATE SOLUTION

HARING

Conference SHARK-FV 2016 S H A R K F V

The MOOD Method A posteriori SPH−MOOD code

PAD, DMP PD, ...

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

ri n+1

Ui n+1

MOOD loop GOOD

quadratic linear

DETECTION

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Chain detectors candidate solution

No

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

PAD

BAD

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

The MOOD Method

Yes No

DMP

GOOD

Yes

PD

Yes

GOOD

No

BAD

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Physical Admissibility Detection (PAD) • PAD requires that the candidate solution remains physically admissible: B Positivity of the density

B Positivity of the pressure

B Positivity of the particle volume

superscript ? refers to the candidate solution Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The MOOD Method

ρ?i > 0 p?i > 0

Vi? > 0

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

NOW-HOW on

• In the application of MOOD to finite volume formulations, Vi is constituted of the cells sharing a face (in 3D) or an edge (in 2D) with the control volume i. • In SPH formulations, we define Vi as the set of closest particles coming from different directions.

OLUME

I Discrete Maximum Principle (DMP) • DMP criterion prevents the spurious numerical oscillations from appearing in the vicinity of discontinuities and steep gradients. An admissible candidate solution fulfills the DMP if it satisfies

n n ? n n

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

j∈Vi

min ρj , ρi ≤ ρi ≤ max ρj , ρi

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

INITE

Conference SHARK-FV 2016 S H A R K F V

The MOOD Method

j∈Vi

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Discrete Maximum Principle (DMP) • We split the reconstruction stencil of the particle i in four sub-domains. For each sub-domain, we choose the closest particle to particle i and include it in Vi.

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The MOOD Method

NW

SW

NE

i SE

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Discrete Maximum Principle (DMP) • Since particles are moving, one should check if the candidate position of the particle i leaves Vi. In that case, the DMP detector defined above is no longer valid, and the PPD for particle i is set to zero.

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The MOOD Method

NW

SW i

vi NE SE

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

NOW-HOW on

INITE

OLUME

Conference SHARK-FV 2016 S H A R K F V

• Due to the truncation for real numbers representation on computer, the constant solution is numerically contaminated by a noise which provides artificial extrema of relative order 10−15. • The DMP detector incorrectly interprets the noise as local extrema, leading to a dramatic order reduction while the solution is mainly constant. • To overcome such a drawback, we introduce the Plateau Detector, which releases the DMP criterion when the solution is almost constant. • For a particle i of stencil Vi, the Plateau Detector is activated if ? ?

I Plateau Detector (PD)

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

The MOOD Method

max(|ρi − ρj |) ≤ εP j∈Vi

where εP > 0 is the Plateau Detector threshold. In this work we have set εP = 10−15.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

•

Introduction

•

The SPH-ALE method

•

MLS approximations

•

The Multi-dimensional Optimal Order Detection (MOOD) Method

•

Application to Euler equations

•

Conclusions

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Outline

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I 2D Ringleb problem • The two-dimensional problem provides a smooth solution we use to measure the effective order of accuracy of the tested SPH scheme.

Universidade do Minho Portugal

LOCAL

COMMITEES:

10

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

log(Lent ) Christophe Berthon | Université de Nantes, France 2

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

10

10

10

10

-3

-5

-7

-9

-11

1000 Base scheme Linear reconstruction Quadratic reconstruction Cubic reconstruction

10000 log(N) 100000

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

• We use 200 particles and solutions are advanced up to t = 0.2. • We use a Taylor reconstruction until the third derivative controlled by the MOOD procedure, with Rusanov numerical flux.

INITE

OLUME

I 1D Riemann problems • Test 1: Sod shock tube on [0, 1] characterized by the initial conditions: (ρ, u, p) =

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:



Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

(1, 0, 1), x ≤ 0.5 (0.125, 0, 0.1), otherwise

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

0

0.4

0.2

0 0

INITE

OLUME

1

0.9

0.8

0.7

ρ0.6

0.5

0.4

0.3

0.2

0.1

0.2

x

1

0.8

p0.6

0.2 0.4

x 0.6

LOCAL Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Exact SPH base scheme SPH−MOOD−MLS

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

0.6

Universidade do Minho Portugal

SCIENTIFIC

0.4

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

0.8

Exact SPH base scheme SPH−MOOD−MLS

0.8 1

1

1 0

2.8

1.6 0

Exact SPH base scheme SPH−MOOD−MLS

0.8

u0.6 0.4

0.2

0 0.2

0.2

0.4

x

0.4

0.6

x 0.6

0.8

0.8

˜ a — Group of Numerical Methods in Engineering Universidade da Corun 1

3 Exact SPH base scheme SPH−MOOD−MLS

2.6

E

2.4

2.2 2

1.8

1

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

INITE

OLUME

ρ

0

0.4

0.2

0 0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.2

x

1

0.8

p0.6

0.2 0.4

x 0.6

LOCAL Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Exact N=800 N=400 N=200 N=100

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

0.6

Universidade do Minho Portugal

SCIENTIFIC

0.4

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

0.8

0.8 1

1

0.8

1 0

Exact N=800 N=400 N=200 N=100

2.8

2.6

1.6 0

Exact N=800 N=400 N=200 N=100

u0.6 0.4

0.2

0 0.2

0.2

0.4

x 0.6

0.4 0.6

0.8

x 0.8

˜ a — Group of Numerical Methods in Engineering Universidade da Corun 1

3 Exact N=800 N=400 N=200 N=100

e 2.4

2.2

2

1.8

1

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

• We use 400 particles where the particles are distributed in two zones. In the first one (covering the domain [−0.6, 0]) we put 320 equally distributed particles while in the second one (domain [0, 0.6]) we place 80 particles. Solutions are advanced up to t = 0.2. • We use a Taylor reconstruction until the third derivative controlled by the MOOD procedure, with Rusanov numerical flux.

INITE

OLUME

I 1D Riemann problems • Test 2: Sod shock tube on Ω = [−0.6, 0.6] characterized by the initial conditions: (ρ, u, p) =

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:



Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

(1, 0, 1), x ≤ 0.5 (0.25, 0, 0.1795), otherwise

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

INITE

OLUME

ρ

0.2

p

0.4

0.2

0

1.1 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

−0.3

−0.3 −0.2

−0.2 −0.1

x

1

0.8

0.6

−0.1 0

x 0.1

LOCAL

Exact SPH base scheme SPH−MOOD−MLS AV−SPH

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

0.1

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

SCIENTIFIC

0

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

0.2

0.2

0.3

Exact SPH base scheme SPH−MOOD−MLS AV−SPH

0.3 0.4

0.9

0.8

0.4

−0.1

1.7

Exact SPH base scheme SPH−MOOD−MLS AV−SPH

0.7

0.6

u 0.5 0.4

0.3

0.2

0.1 0 −0.3

−0.3

−0.2

−0.2

−0.1

−0.1

0

2.7

x

0

x

0.1

2.6

2.5

0.1

0.2

0.2

0.3

0.3

˜ a — Group of Numerical Methods in Engineering Universidade da Corun 0.4

Exact SPH base scheme SPH−MOOD−MLS AV−SPH

2.4

E

2.3

2.2

2.1 2

1.9

1.8

0.4

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Circular blast wave problems • The computational domain is an open disk with center in (0, 0) and radius R = 1.5. • Two configurations: A regular and radially symmetric and a unstructured set of 74651 particles.

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Circular blast wave problems • The computational domain is an open disk with center in (0, 0) and radius R = 1.5. • Two configurations: A regular and radially symmetric and a unstructured set of 74651 particles.

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Circular blast wave problems • For the first blast wave problem case, the initial condition is (ρ, u, v, p) =

• Solutions are advanced up to t = 0.2. • We use a Taylor reconstruction until the third derivative controlled by the MOOD procedure, with Rusanov numerical flux. Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:



Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

(1, 0, 0, 1), if |r| ≤ 0.5 (0.125, 0, 0, 0.1), otherwise

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Circular blast wave problems • Density distribution at t = 0.2

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

0 −1

0.4

0.2

0 −1

INITE

OLUME

1

0.8

ρ0.6

0.4

0.2 Exact SPH Base scheme SPH−MOOD−MLS

−0.5

x

1

0.8

p0.6

−0.5 0

x

Exact SPH Base scheme SPH−MOOD−MLS

LOCAL

0.5 Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

0

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

1

0.8

ρ0.6

0.4

0.2

0.5

1

1 0 −1

0 −1

Exact SPH Base scheme SPH−MOOD−MLS

−0.5

x

0.2

−0.5

0

0

0.5

x

0.5

˜ a — Group of Numerical Methods in Engineering Universidade da Corun 1

1

0.8

p0.6

0.4

Exact SPH Base scheme SPH−MOOD−MLS 1

IGHER-ORDER

DVANCED

ESEARCH

Density

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

0.6

0.5

0.4

0.3

Density

INITE

OLUME

•

0

0

Density (left panels), norm of the velocity (middle panels) and pressure (right panels) as a function of

particle radius for all particles.

1

0.9 Density - SPH-MOOD - Structured particles

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1 0.2

0.2 0.4 Radius

1

0.9 Density - SPH-MOOD - Unstructured particles

0.8

0.7

0.4 Radius 0.6 0.8

LOCAL

0.1 1

Stéphane Clain |Velocity| Jorge Figueiredo Gaspar Machado Rui Pereira

1

0

0.6

0 Pressure

1

0.8

0.6

0.4

Pressure

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy |Velocity| de Toulouse, France Raphaël Loubère | Université Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

0.8

Universidade do Minho Portugal

0.6

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

|V| - SPH-MOOD - Structured particles

0.2

0.2

0.4 0.6

0.4 Radius 0.6

0.8

0.8

1

1

|V| - SPH-MOOD - Unstructured particles

0.8

0.4

1

Pressure - SPH-MOOD - Structured particles

0.8

0.6

0.4

0.2

0.2

0

Radius 0

0

0.2

0.2

0.4 Radius

1 0.6

0.4 Radius 0.6

˜ a — Group of Numerical Methods in Engineering Universidade da Corun 0.8

0.8

1

1

Pressure - SPH-MOOD - Unstructured particles

0.8

0.6

0.4

0.2 0.2 0.2

0 1

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Circular blast wave problems • For the second blast wave problem case, the initial condition is (ρ, u, v, p) =

• Solutions are advanced up to t = 0.2. • We use a Taylor reconstruction until the third derivative controlled by the MOOD procedure, with Rusanov numerical flux. Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

 (1, 0, 0, 2), if |r| ≤ 0.5 (1, 0, 0, 1), otherwise

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I Circular blast wave problems • Density distribution at t = 0.2

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

INITE

OLUME

ρ

0.6 −1

1.5

−1

2

1.8

1.6

1.4

1.2

1

0.8

−0.5

x

2.5

2

p

−0.5

x 0

LOCAL 0.5 Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

Exact SPH Base scheme SPH−MOOD−MLS

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

0

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

2

0.5

1

1

1.8

ρ

1 0.6 −1

−1

Exact SPH Base scheme SPH−MOOD−MLS

1.6

1.4

1.2

1

0.8

−0.5

x

−0.5

0

Exact SPH Base scheme SPH−MOOD−MLS

0 0.5

x

0.5

˜ a — Group of Numerical Methods in Engineering Universidade da Corun 1

2.5 Exact SPH Base scheme SPH−MOOD−MLS

2

p 1.5

1

1

IGHER-ORDER

DVANCED

ESEARCH

Density

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

1

0.9

Density

INITE

OLUME

•

0

0

Density (left panels), norm of the velocity (middle panels) and pressure (right panels) as a function of

particle radius for all particles.

1.3 Density - SPH-MOOD - Structured particles

1.2

1.1

1

0.9

0.8

0.7 0.2

0.2 0.4 Radius

1.3 Density - SPH-MOOD - Unstructured particles

1.2

1.1

0.4 0.6

Radius

0.8

0.8

LOCAL

0.7 1

Stéphane Clain |Velocity| Jorge Figueiredo Gaspar Machado Rui Pereira

1

0

0.25

0.2

0 Pressure

0.4

0.35

0.3

0.25

0.2

0.15

Pressure

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy |Velocity| de Toulouse, France Raphaël Loubère | Université Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

0.8

Universidade do Minho Portugal

0.6

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

0.45

|V| - SPH-MOOD - Structured particles

0.2

0.2

0.4

0.45

0.6

0.4 0.6

Radius

0.8

0.8

2

0.1

0

1

|V| - SPH-MOOD - Unstructured particles

0.35

0.1

0 1

Pressure - SPH-MOOD - Structured particles

1.8

1.6

1.4

1.2

0.05

1

Radius 0

0

0.2

0.2

0.4 Radius

2 0.6

0.4 Radius 0.6

˜ a — Group of Numerical Methods in Engineering Universidade da Corun 0.8

0.8

1

0.4

Pressure - SPH-MOOD - Unstructured particles

1.8

0.3

1.6

0.15 1.4

1.2

0.05

1 1

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I 1D Riemann problems • Test: Very strong blast shock wave Ω = [−0.5, 0.5] characterized by the initial conditions:

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

(ρ, u, p) =

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

 (1, 0, 1000), x≤0 (1, 0, 0.01), otherwise

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

7

6

5

4

ρ

3

2

1

0 0.1 0.11

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

0.12 0.13 0.14 0.15 x 0.16 0.17 0.18 0.19 0.2

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I The problem is related to the fact that the condition

is not verified I We modify the scheme for variable hi to verify the previous equation and Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

N X

Vj Wij = 1

j=1

hi = σ(Vi)1/D

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

ni X dViU i Gij − H i) · ∇Wij = − ViVj 2 (G dt j=1

I The equation for Vi acts as a predictor dVi = dt

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

The SPH-ALE method

ni X

ViVj 2 (vv ij − v i) ∇Wij

j=1

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

• We use the following algorithm for variable hi

B 1) We use the values at time n (U U n,V n,hn) to solve the continuity, momentum and energy equations. B 2) We obtain a prediction of the weights using the equation for Vi. B 3) Loop to impose partition of unity ◦ 3.1) We obtain h∗i from the derivative of hVi = σ(Vi)1/D .

◦ 3.2) We compute Vj using h∗i

◦ 3.3) With this Vi we recompute hVi ◦ 3.4) Proceed until |h∗i − hVi |/h∗i is less than a given tolerance (10−2, less than 10 iterations). Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

σ hVi d(Vi) dhi = ⇒ h∗i = hni + ∆t · dt D Vi dt

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

The SPH-ALE method

N X

j=1



ViWij = 1 ⇒ Vi = 

N X

 σ hVi d(Vi) D Vi dt

˜ a — Group of Numerical Methods in Engineering Universidade da Corun



(−1)

Wij 

j=1

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

7

6

5

4

ρ

3

2

1

0 0.1 0.11

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Application to Euler equations

0.12 0.13 0.14 0.15 x 0.16 0.17 0.18 0.19 0.2

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

•

Introduction

•

The SPH-ALE method

•

MLS approximations

•

The Multi-dimensional Optimal Order Detection (MOOD) Method

•

Application to Euler equations

•

Conclusions

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Outline

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

I Our formulation obtains accurate and very promising results.

INITE

OLUME

I A new high-accurate SPH code has been presented.

I It is based on the use of Riemann solvers and Moving Least Squares for high accurate reconstruction of the left and right states of the Riemann problem.

I The stability of the numerical scheme is achieved by using the Multidimensional Optimal Order Detection (MOOD) paradigm. Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

NOW-HOW on

Conference SHARK-FV 2016 S H A R K F V

Conclusions

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

AN SPH METHOD WITH MOOD LIMITING FOR COMPRESSIBLE FLOWS

Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Xes´ us Nogueira

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

SHARK-FV 2016

email: [email protected]

Thank you —

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IGHER-ORDER

DVANCED

ESEARCH

NOW-HOW on

INITE

OLUME

I This work has been partially supported by: • The Ministerio de Econom´ıa y Competitividad of the Spanish Government, • Direcci´ on Xeral de I+D of the Conseller´ıa de Cultura, Educaci´ on e Ordenaci´ on Universitaria of the Xunta de Galicia, cofinanced with FEDER funds. • the Universidade da Coru˜ na (UDC), and • the Group of Numerical Methods in Engineering - GMNI Stéphane Clain Jorge Figueiredo Gaspar Machado Rui Pereira

Universidade do Minho Portugal

LOCAL

COMMITEES:

Christophe Berthon | Université de Nantes, France Stéphane Clain | Universidade do Minho, Portugal Frédéric Coquel | Université Pierre et Marie Curie, France Steven Diot | CEA, France Michael Dumbser | Università degli studi di Trento, Italy Raphaël Loubère | Université de Toulouse, France Elena Vázquez Cendón| Universidade de Santiago de Compostela, Spain Xesús Nogueira| Universidade da Coruña, Spain

SCIENTIFIC

High-Order Scheme Derivations: hyperbolic, parabolic, elliptic Non-Conservative Hyperbolic Systems of PDE's with Stiff Sources High Performance Computing on new Architectures Multispecies/Multimaterial Fluid Flows Robustness and Stability Properties Biomedical application of High-Order Schemes

THEMES:

Ofir, Portugal May 23 - 27, 2016

Great White

www.math.univ-toulouse.fr/SHARK-FV

HARING

Conference SHARK-FV 2016 S H A R K F V

Acknowledgments

˜ a — Group of Numerical Methods in Engineering Universidade da Corun