A Direct Arbitrary-Lagrangian-Eulerian ADER-WENO Finite Volume Scheme on Unstructured Tetrahedral Meshes for Conservative and Nonconservative Hyperbolic Systems in 3D Walter Boscheri and Michael Dumbser [email protected] University of Trento (Italy) - Department of Civil, Environmental and Mechanical Engineering -

SHARK FV 2015 May, 18th − 22nd 2015 - Ofir

Outline 1

Introduction

2

Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm

3

Validation The Euler equations The Baer–Nunziato Model

4

Conclusions and Outlook

1. Introduction

Introduction: Lagrangian methods D ∂ () = () + v ∇() Dt ∂t Advantages availability of trajectory information; no convection terms → numerical stability; less numerical diffusion; material interfaces are precisely located and identified.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

2 / 19

1. Introduction

Introduction: Lagrangian methods D ∂ () = () + v ∇() Dt ∂t Advantages availability of trajectory information; no convection terms → numerical stability; less numerical diffusion; material interfaces are precisely located and identified.

Disadvantages high computational cost; mesh distortion. Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

2 / 19

2. Numerical Method

Hyperbolic Balance Laws

Outline 1 2

3

4

Introduction Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm Validation The Euler equations The Baer–Nunziato Model Conclusions and Outlook

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

2 / 19

2. Numerical Method

Hyperbolic Balance Laws

Hyperbolic Balance Laws ∂Q + ∇ · F(Q) + B(Q) · ∇Q = S(Q), ∂t

x ∈ Ω ⊂ R3 , t ∈ R+ 0

(1)

where Q ∈ ΩQ ⊂ Rν

→

state vector

F = (f, g, h)

→

flux tensor (purely conservative part of (1))

B = (B1 , B2 , B3 )

→

non–conservative terms of (1)

S(Q)

→

vector of algebraic source terms

P(Q) = B(Q) · ∇Q

→

abbreviation

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

3 / 19

2. Numerical Method

Hyperbolic Balance Laws

Hyperbolic Balance Laws Moving mesh TΩn =

NE [

Tin

i=1

k = 1, 2, 3, 4

∀Tin

x

=

(x, y , z)

ξ

=

(ξ, η, ζ)

Walter Boscheri

x

=

y

=

z

=

      n n n n n n n X1,i + X2,i − X1,i ξ + X3,i − X1,i η + X4,i − X1,i ζ       n n n n n n n Y1,i + Y2,i − Y1,i ξ + Y3,i − Y1,i η + Y4,i − Y1,i ζ       n n n n n n n Z1,i + Z2,i − Z1,i ξ + Z3,i − Z1,i η + Z4,i − Z1,i ζ High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

3 / 19

2. Numerical Method

Hyperbolic Balance Laws

Hyperbolic Balance Laws Moving mesh TΩn =

NE [

Tin

i=1

k = 1, 2, 3, 4

∀Tin

x

=

(x, y , z)

ξ

=

(ξ, η, ζ)

Data representation Qni =

Walter Boscheri

1 |Tin |

Z

Q(x, y , z, t n )dV

Tin

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

3 / 19

2. Numerical Method

High Order WENO Reconstruction on Unstructured Meshes

Outline 1 2

3

4

Introduction Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm Validation The Euler equations The Baer–Nunziato Model Conclusions and Outlook

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

3 / 19

2. Numerical Method

High Order WENO Reconstruction on Unstructured Meshes

High Order of Accuracy in Space Some spatial basis functions ψl (ξ, η, ζ) are used to write the reconstructed polynomials whs for each reconstruction stencil s: whs (x, y , z, t n ) =

M X

n,s n,s ˆ l,i ˆ l,i ψl (ξ, η, ζ)w := ψl (ξ, η, ζ)w

l=1

Integral conservation

1 |Tjn |

Z

ne [

Sis

=

n Tm(j)

M

=

(M + 1)(M + 2)(M + 3)/6

ne

=

3M

j=1 n,s ˆ l,i ψl (ξ, η, ζ)w dV = Qnj ,

Tjn

Walter Boscheri

∀Tjn ∈ Sis

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

4 / 19

2. Numerical Method

High Order WENO Reconstruction on Unstructured Meshes

High Order of Accuracy in Space Some spatial basis functions ψl (ξ, η, ζ) are used to write the reconstructed polynomials whs for each reconstruction stencil s: whs (x, y , z, t n ) =

M X

n,s n,s ˆ l,i ˆ l,i ψl (ξ, η, ζ)w := ψl (ξ, η, ζ)w

l=1

Integral conservation

1 |Tjn |

Z

ne [

Sis

=

n Tm(j)

M

=

(M + 1)(M + 2)(M + 3)/6

ne

=

3M

j=1 n,s ˆ l,i ψl (ξ, η, ζ)w dV = Qnj ,

∀Tjn ∈ Sis

Tjn

WENO polynomials wh (x, y , z, t n ) =

M X

n ˆ l,i ψl (ξ, η, ζ)w ,

with

n ˆ l,i w =

X

nonlinear WENO weights

oscillation indicators 

ω ˜s =

λs (σs + )r

Walter Boscheri

,

ω ˜s ωs = P ˜q qω

n,s ˆ l,i ωs w

s

l=1

 σs = 

X

Z

α+β+γ≤M T e

∂ α+β+γ ψl (ξ, η, ζ) ∂ξα ∂η β ∂ζ γ

·

∂ α+β+γ ψm (ξ, η, ζ) ∂ξα ∂η β ∂ζ γ

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes



 dξdηdζ · 4 / 19

2. Numerical Method

Local Space-Time DG Predictor on Moving Meshes

Outline 1 2

3

4

Introduction Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm Validation The Euler equations The Baer–Nunziato Model Conclusions and Outlook

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

4 / 19

2. Numerical Method

Local Space-Time DG Predictor on Moving Meshes

High Order of Accuracy in Time ˜ x = (x, y , z, t) x = (x, y , z) physical coordinates

ξ˜ = (ξ, η, ζ, τ ) ξ = (ξ, η, ζ) reference coordinates

˜ = θl (ξ, η, ζ, τ ) θl = θl (ξ) space–time basis functions

Isoparametric approach qh = qh (ξ, η, ζ, τ ) = θl (ξ, η, ζ, τ )b ql,i

bl,i Fh = (fh , gh , hh ) = Fh (ξ, η, ζ, τ ) = θl (ξ, η, ζ, τ )F

b l,i ˜ = θl (ξ) ˜ P Ph = B(qh ) · ∇qh = Ph (ξ)

bl,i Sh = Sh (ξ, η, ζ, τ ) = θl (ξ, η, ζ, τ )S ⇓

x(ξ, η, ζ, τ ) = θl (ξ, η, ζ, τ )b xl,i , z(ξ, η, ζ, τ ) = θl (ξ, η, ζ, τ )b zl,i ,

y (ξ, η, ζ, τ ) = θl (ξ, η, ζ, τ )b yl,i t(ξ, η, ζ, τ ) = θl (ξ, η, ζ, τ )tbl

Jacobian 

xξ  yξ ∂˜ x  Jst = = ∂ ξ˜  zξ 0 Walter Boscheri

xη yη zη 0

xζ yζ zζ 0

 xτ yτ   zτ  ∆t



Jst−1

ξx ∂ ξ˜  ηx  = = ζx ∂˜ x 0

ξy ηy ζy 0

ξz ηz ζz 0

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

 ξt ηt   ζt 

1 ∆t

5 / 19

2. Numerical Method

Local Space-Time DG Predictor on Moving Meshes

High Order of Accuracy in Time The PDE in the reference coordinate system

  ∇ξ = 

∂ ∂ξ ∂ ∂η ∂ ∂ζ

  ,

 ∇=

∂ ∂x ∂ ∂y ∂ ∂z





ξx  =  ξy ξz

ηx ηy ηz

 ζx  ζy   ζz

∂ ∂ξ ∂ ∂η ∂ ∂ζ

   =

∂ξ ∂x

T ∇ξ

⇓ " #  T  T ∂Q ∂Q ∂ξ ∂ξ ∂ξ + ∆t · + ∇ξ · F + B(Q) · ∇ξ Q = ∆tS(Q) ∂τ ∂ξ ∂t ∂x ∂x Let define the following operators: [f , g ]τ =

Z1 Z

Z f (ξ, η, ζ, τ )g (ξ, η, ζ, τ )dξdηdζ, Te

Walter Boscheri

hf , g i =

f (ξ, η, ζ, τ )g (ξ, η, ζ, τ )dξdηdζdτ. 0 Te

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

5 / 19

2. Numerical Method

Local Space-Time DG Predictor on Moving Meshes

High Order of Accuracy in Time ∂Q = ∆tH ∂τ "

∂Q ∂ξ H = ∆tS − ∆t · + ∂ξ ∂t



∂ξ ∂x

T

 ∇ξ · F + B(Q) ·

∂ξ ∂x

#

T

∇ξ Q ,

b l,i . ˜ H Hh = θl (ξ)

Multiplication with θk (ξ) and integration over Te × [0, 1] yields the weak formulation of the PDE, i.e. 

Walter Boscheri

∂θl θk , ∂τ

 b l,i bl,i = hθk , θl i ∆t H q

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

5 / 19

2. Numerical Method

Local Space-Time DG Predictor on Moving Meshes

High Order of Accuracy in Time The nonlinear algebraic equation system can be written in a more compact matrix form as n b l,i bl,i = F0 w ˆ l,i + MH K1 q

1

K1 = [θk (ξ, 1), θl (ξ, 1)] −

Walter Boscheri



∂θk , θl ∂τ

 ,

F0 = [θk (ξ, 0), ψl (ξ)] ,

M = hθk , θl i .

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

5 / 19

2. Numerical Method

Local Space-Time DG Predictor on Moving Meshes

High Order of Accuracy in Time The nonlinear algebraic equation system can be written in a more compact matrix form as n b l,i bl,i = F0 w ˆ l,i + MH K1 q

1

K1 = [θk (ξ, 1), θl (ξ, 1)] −



∂θk , θl ∂τ

 ,

F0 = [θk (ξ, 0), ψl (ξ)] ,

M = hθk , θl i .

local PDE evolution   n br brl,i+1 = K1−1 F0 w ˆ l,i q + MH l,i

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

5 / 19

2. Numerical Method

Local Space-Time DG Predictor on Moving Meshes

High Order of Accuracy in Time The nonlinear algebraic equation system can be written in a more compact matrix form as n b l,i bl,i = F0 w ˆ l,i + MH K1 q

1

K1 = [θk (ξ, 1), θl (ξ, 1)] −



∂θk , θl ∂τ

local PDE evolution   n br brl,i+1 = K1−1 F0 w ˆ l,i q + MH l,i

 ,

F0 = [θk (ξ, 0), ψl (ξ)] ,

M = hθk , θl i .

geometry evolution dx = V (Q, x, t) dt

b l,i Vh = θl (ξ, τ )V b l,i = V(ˆ V ql,i , ˆxl,i , tˆl )

0 b l,i K1 b xl,i = [θk (ξ, 0), x(ξ, t n )] + ∆tM V Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

5 / 19

2. Numerical Method

Mesh Motion

Outline 1 2

3

4

Introduction Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm Validation The Euler equations The Baer–Nunziato Model Conclusions and Outlook

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

5 / 19

2. Numerical Method

Mesh Motion

Nodal solvers 1) The node solver of Cheng and Shu N S cs (JCP 227: 1567-1596, 2007)  1  Z 1 X n n n b l,j Vk = Vk,j with Vk,j =  θl (ξe,m(k) , ηe,m(k) , ζe,m(k) , τ )dτ  V µk j∈V k

0

X

µk =

1 = Nk

Tjn ∈Vk

or µk =

X

µk,j

µk,j = ρnj |Tjn |

Tjn ∈Vk

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

6 / 19

2. Numerical Method

Mesh Motion

Nodal solvers 2) The node solver of Maire N S m n

(JCP 228: 2391-2425, 2009)

Vk = Mk−1

X

(Lk,j Pj nj + Mp,j Vj )

j∈Vk

with Mk,j =

X f ∈j

Walter Boscheri

  j Zk,f Ljk,f njk,f ⊗ njk,f ,

Mk =

X

Mk,j ,

j Zk,f = ρj cj

j∈Vk

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

6 / 19

2. Numerical Method

Mesh Motion

Geometry Evolution Lagrangian coordinates n

xn+1,Lag = xnk + Vk · ∆t k Rezoned coordinates 

Fk (xk ) =

X j∈Vk

K(Jj ),

x j − xkj  k+3 j Jj =  yk+3 − ykj j zk+3 − zkj xrez k

→

 j xk+1 − xkj  j yk+1 − ykj  j zk+1 − zkj

min {Fk (xk )}

New coordinates (relaxation algorithm 1 )   n+1,Lag n+1,Lag rez + ω x − x , xn+1 = x k k k k k 1

j xk+2 − xkj j yk+2 − ykj j zk+2 − zkj

  ωk = F xnk , xn+1,Lag k

Galera et al., JCP 229: 5755-5787, 2010

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

7 / 19

2. Numerical Method

Solution Algorithm

Outline 1 2

3

4

Introduction Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm Validation The Euler equations The Baer–Nunziato Model Conclusions and Outlook

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

7 / 19

2. Numerical Method

Solution Algorithm

Finite Volume Scheme The starting PDE system can be rewritten as ˜ + B(Q) ˜ ˜ ·F ˜ = S(Q) ∇ · ∇Q with ˜ = ∇



∂ ∂ ∂ ∂ , , , ∂x ∂y ∂z ∂t

T ,

˜ = (f, g, h, Q) , F

˜ = (B1 , B2 , B3 ). B

  Integration over the space–time control volume Cin = Ti (t) × t n ; t n+1 yields Z Cin

Walter Boscheri

˜ dxdt + ˜ ·F ∇

Z Cin

˜ ˜ dxdt = B(Q) · ∇Q

Z S(Q) dxdt Cin

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

8 / 19

2. Numerical Method

Solution Algorithm

Finite Volume Scheme Application of Gauss’ theorem leads to Z Z  Z  ˜+D ˜ · n ˜ ˜ dxdt = S(Q) dxdt F ˜ dS + B(Q) · ∇Q ∂Cin

Cin

Cin \∂Cin

where n ˜ = (˜ nx , n˜y , n˜z , n˜t ),

1 ˜ ·n D ˜= 2

Z1


∂Ψ ˜ Ψ(Q− , Q+ , s) · n B ˜ ds. ∂s

0

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

8 / 19

2. Numerical Method

Solution Algorithm

Finite Volume Scheme Application of Gauss’ theorem leads to Z Z  Z  ˜+D ˜ · n ˜ ˜ dxdt = S(Q) dxdt F ˜ dS + B(Q) · ∇Q ∂Cin

Cin

Cin \∂Cin

where n ˜ = (˜ nx , n˜y , n˜z , n˜t ),

1 ˜ ·n D ˜= 2

Z1


∂Ψ ˜ Ψ(Q− , Q+ , s) · n B ˜ ds. ∂s

0

Linear parametrization ∂Cijn = ˜ x (χ1 , χ2 , τ ) =

6 X

˜ nij,k βk (χ1 , χ2 , τ )X

k=1

0≤χ ˜ = (χ1 , χ2 , τ ) ≤ 1 |∂Cijn | Walter Boscheri

∂˜ x = , ∂χ ˜

n ˜ij =

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

∂˜ x /|∂Cijn | ∂χ ˜ 8 / 19

2. Numerical Method

Solution Algorithm

Finite Volume Scheme ALE–type One–Step Finite Volume Scheme |Tin+1 | Qn+1 i

=

|Tin | Qni

−

1 1 X Z Z Tj ∈Ni

Walter Boscheri

0

0

˜ ij dχdτ + |∂Cijn |G

Z (Sh − Ph ) dxdt

Cin \∂Cin

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

8 / 19

2. Numerical Method

Solution Algorithm

Finite Volume Scheme ALE–type One–Step Finite Volume Scheme |Tin+1 | Qn+1 i

=

|Tin | Qni

1 1 X Z Z

−

Tj ∈Ni

0

˜ ij dχdτ + |∂Cijn |G

Z (Sh − Ph ) dxdt

Cin \∂Cin

0

Rusanov numerical flux  1  Z   1 −
˜ ij = 1 F(q ˜ + ˜ − ˜ ˜ij +  B(Ψ) G ·n ˜ ds − |λmax |I q+ h ) + F(qh ) · n h − qh 2 2 0

˜ n˜ = A ˜ ·n A ˜=

Walter Boscheri

q

  n˜x2 + n˜y2 + n˜z2

  ∂F + B · n − (V · n) I , ∂Q

(˜ nx , n˜y , n˜z )T n= p n˜x2 + n˜y2 + n˜z2

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

8 / 19

2. Numerical Method

Solution Algorithm

Finite Volume Scheme ALE–type One–Step Finite Volume Scheme |Tin+1 | Qn+1 i

=

|Tin | Qni

−

1 1 X Z Z Tj ∈Ni

0

˜ ij dχdτ + |∂Cijn |G

Z (Sh − Ph ) dxdt

Cin \∂Cin

0

Osher numerical flux   1 Z     1 ˜ + −
˜ ij = 1 F(q ˜ + ˜ − ˜ ˜ij +  G B(Ψ) ·n ˜ − A n ˜ (Ψ) ds  qh − qh h ) + F(qh ) · n 2 2 0

Ψ = Ψ(Q− , Q+ , s) = Q− + s(Q+ − Q− ), |A| = R|Λ|R−1 , Walter Boscheri

0≤s≤1

|Λ| = diag (|λ1 |, |λ2 |, ..., |λν |)

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

8 / 19

3. Validation

The Euler equations

Outline 1 2

3

4

Introduction Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm Validation The Euler equations The Baer–Nunziato Model Conclusions and Outlook

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

8 / 19

3. Validation

The Euler equations

The Euler equations

Qt + fx + gy + hz = S(Q) with



 ρ Q =  ρvj  , ρE



 ρvi F = (f, g, h) =  ρvi vj + pδij  . vi (ρE + p)

ρ → fluid density v = (vx , vy , vz ) → fluid’s velocity vector S → vector of source term (S = 0) p → fluid pressure

Walter Boscheri

  1 p = (γ−1) ρE − ρ(vx2 + vy2 + vz2 ) 2 ↓ equation of state

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

9 / 19

3. Validation

The Euler equations

Convergence studies: Shu Vortex (Hu and Shu, JCP 150: 97-127 , 1999 )

IC: (ρ, vx , vy , vz , p) = (1 + δρ, 1 + δvx , 1 + δvy , 0, 1 + δp) Computational domain Ω(0) = [0; 10] × [0; 10] × [0; 5] Convective velocity vc = (1, 1, 0) Perturbations 

 δvx  δvy  δvz

=

δS

=

δT Walter Boscheri

=

Parameters   −(y − 5)  1−r2 2  (x − 5)  e 2π 0 0 −

(γ − 1)

r2

=

(x − 5)2 + (y − 5)2 radius



=

5 vortex strength

γ

=

1.4 ratio of specific heat

2

2 e 1−r High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

10 / 19

3. Validation

The Euler equations

Convergence studies: Shu Vortex (Hu and Shu, JCP 150: 97-127 , 1999 ) h(Ω(tf )) 3.43E-01 2.85E-01 2.09E-01 1.47E-01 2.89E-01 2.17E-01 1.52E-01 1.13E-01 2.89E-01 2.17E-01 1.52E-01 1.13E-01 Walter Boscheri

L 2 O1 1.081E-01 9.159E-02 6.875E-02 4.899E-02 O3 1.718E-02 7.641E-03 2.601E-03 1.049E-03 O5 2.272E-03 6.605E-04 1.234E-04 2.932E-05

O(L2 )

h(Ω, tf )

0.9 0.9 1.0

2.89E-01 2.16E-01 1.52E-01 1.13E-01

2.8 3.1 3.1

2.89E-01 2.17E-01 1.52E-01 1.13E-01

4.3 4.8 4.9

2.89E-01 2.17E-01 1.52E-01 1.13E-01

L 2 O2 2.214E-02 1.202E-02 5.865E-03 3.254E-03 O4 4.116E-03 1.369E-03 3.273E-04 9.802E-05 O6 1.015E-03 2.312E-04 3.090E-05 6.576E-06

O(L2 ) 2.1 2.0 2.0 3.8 4.1 4.1 5.1 5.7 5.2

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

10 / 19

3. Validation

The Euler equations

Three–dimensional Explosion Problem Inner state (Qi ) Outer state (Qo )

ρ 1.0 0.125

vx 0.0 0.0

vy 0.0 0.0

vz 0.0 0.0

p 1.0 0.1

Parameters r2

=

x 2 + y 2 + z 2 radius

R

=

0.5 internal radius

γ

=

1.4 ratio of specific heat

Analytical solution Qt + F(Q)r = S(Q) 

 ρ Q =  ρu  , ρE Walter Boscheri



 ρu F =  ρu + p  , u(ρE + p) 2

 α S=− r

 ρu 2  ρu u(ρE + p)

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

α=2

11 / 19

3. Validation

The Euler equations

Three–dimensional Explosion Problem

t = 0.00

t = 0.08

4th order accurate explosion problem with NE = 7225720. Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

11 / 19

3. Validation

The Euler equations

Three–dimensional Explosion Problem

t = 0.16

t = 0.25

4th order accurate explosion problem with NE = 7225720. Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

11 / 19

3. Validation

The Euler equations

Three–dimensional Explosion Problem Density profile Reference solution ALE WENO (O4)

1.3 1.2 1.1 1 0.9 0.8

rho

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

4th order accurate explosion problem with NE = 7225720. Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

11 / 19

3. Validation

The Euler equations

Three–dimensional Explosion Problem Velocity and pressure profile Reference solution ALE WENO (O4)

1.2

1.1

1.1

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 -0.1

Reference solution ALE WENO (O4)

1.3

1.2

p

u

1.3

0 0

0.1

0.2

0.3

0.4

0.5

x

0.6

0.7

0.8

0.9

1

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

4th order accurate explosion problem with NE = 7225720. Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

11 / 19

3. Validation

The Kidder problem IC:

The Euler equations

(Kidder, Nucl. Fusion 1: 3-14 , 1976 )

ρ0 = ρ(r , 0) =

2 − r2 re,0 2 − r2 re,0 i,0

γ−1 ρi,0 +

!

2 r 2 − ri,0 2 − r2 re,0 e,0

1 γ−1

ργ−1 e,0

p0 =1 ργ0 p0 (r ) = s0 ρ0 (r )γ s0 =

Z Y X

Parameters 1

re (0) = 1.0 external radius

0.8

ri (0) = 0.9 internal radius

0.6

z

γ = 5/3 ratio of specific heat ρi,0 = 1.0 initial internal density

0.4

1

0.2 0.8 0.6

y

0

ρe,0 = 2.0 initial external density

0.4 0

0.2

0.2 0.4

0.6

x

Walter Boscheri

0.8

0 1

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

12 / 19

3. Validation

The Euler equations

The Kidder problem

(Kidder, Nucl. Fusion 1: 3-14 , 1976 )

Analytical Solution 

 R(r , t) ρ0 ρ (R(r , t), t) = h(t) h(t)   R(r , t) d h(t) ur (R(r , t), t) = dt h(t)   2γ R(r , t) p (R(r , t), t) = h(t)− p0 γ−1 h(t) 2 − γ−1

with r h(t) =

t2 1− 2, τ

s τ =

2

2

γ − 1 (re,0 − ri,0 ) 2 2 2 ce,0 − ci,0

and the internal and external sound speeds ci and ce defined as r r pi pe ci = γ , ce = γ ρi ρe Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

12 / 19

3. Validation

The Kidder problem

The Euler equations

(Kidder, Nucl. Fusion 1: 3-14 , 1976 )

Rinternal: exact solution Rinternal: ALE WENO (O4) Rexternal: exact solution Rexternal: ALE WENO (O4)

1.1

1

Radius

0.9

0.8

0.7

0.6

0.5

0.4 -0.05

0

0.05

0.1

0.15

0.2

0.25

time

Internal radius External radius Walter Boscheri

rex 0.450000 0.500000

rnum 0.449765 0.499728

|err | 2.35E-04 2.72E-04

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

12 / 19

3. Validation

The Euler equations

The Saltzman problem

(Dukovicz et al., JCP 99: 115-134 , 1992 ) Z Y X

Computational domain Ω(0) = [0; 1] × [0; 0.1] × [0; 0.1]

0

Piston velocity vp = (1, 0, 0)

0.2

0.4

x

0.6

0.1

0.08

z

0.06

0.8

0.04

0.02

0 0.1 0.08 0.06 1 0.04

y

0.02 0

Z Y

Mesh size 100 × 10 × 10 cubes

X

0

IC: ρ0 = 1, v0 = 0, e0 = 10−4

0.2

0.4

x

0.6

0.1

0.08

z

0.06

0.8

0.04

0.02

0 0.1 0.08 0.06 1 0.04

y

0.02 0

Analytical solution ρ u p

Left state 1.0 1.0 6.67 · 10−5

Walter Boscheri

Right state 1.0 -1.0 6.67 · 10−5

final time: shifting: final shock location: exact density:

tf = 0.6 d = up · tf x = 0.8 ρe = 4.0

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

13 / 19

3. Validation

The Saltzman problem

Walter Boscheri

The Euler equations

(Dukovicz et al., JCP 99: 115-134 , 1992 )

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

13 / 19

3. Validation

The Euler equations

The Saltzman problem

(Dukovicz et al., JCP 99: 115-134 , 1992 )

Exact solution ALE WENO (O3)

4.5

Exact solution ALE WENO (O3)

1.4 1.2

4

1

3.5

0.8 3

u

rho

0.6 2.5

0.4 2 0.2 1.5

0

1 0.5 0.6

-0.2

0.65

0.7

0.75

0.8

x

0.85

0.9

0.95

1

-0.4 0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

x

3rd order numerical results of density and velocity for the Saltzman problem.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

13 / 19

3. Validation

The Euler equations

The Sedov Problem Computational domain Ω(0) = [0; 1.2] × [0; 1.2] × [0; 1.2] Mesh size 40 × 40 × 40 cubes IC: ρ0 = 1, v0 = 0, γ = 1.4, Etot = 0.851072  p0 =

Etot (γ − 1)ρ0 8·V or −6 10

in cor , elsewhere.

Analytical solution ⇒ Kamm et al., Technical Report LA-UR-07-2849, LANL, 2007 Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

14 / 19

3. Validation

The Euler equations

The Sedov Problem

Exact solution ALE WENO (O3)

6

5

4

rho

3

2

1

0

-1

0

0.2

0.4

0.6

0.8

1

1.2

r

3rd order accurate numerical results for the Sedov problem.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

14 / 19

3. Validation

The Euler equations

The Sedov Problem

3rd order accurate numerical results for the Sedov problem.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

14 / 19

3. Validation

The Baer–Nunziato Model

Outline 1 2

3

4

Introduction Numerical Method Hyperbolic Balance Laws High Order WENO Reconstruction on Unstructured Meshes Local Space-Time DG Predictor on Moving Meshes Mesh Motion Solution Algorithm Validation The Euler equations The Baer–Nunziato Model Conclusions and Outlook

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

14 / 19

3. Validation

The Baer–Nunziato Model

The Baer-Nunziato Model of Compressible Multi-Phase Flows ∂ ∂t

        ∂   (φ ρ u ) + ∇ · (φ ρ u u ) + ∇φ p = p ∇φ − λ (u − u ) , 1 1 1 1 1 1 1 1 1 1 I 1 2  ∂t       ∂  (φ ρ E ) + ∇ · ((φ ρ E + φ p ) u ) = −p ∂ φ − λ u · (u − u ) , 1 1 1 1 1 1 1 1 t 1  1 I I 1 2 ∂t     ∂ (φ ρ ) + ∇ · (φ ρ u ) = 0, 2 2 2 2 2 ∂t      ∂   (φ ρ u ) + ∇ · (φ ρ u u ) + ∇φ p = p ∇φ − λ (u − u ) , 2 2 2 2 2 2 2 2 2 2 I 2 1  ∂t       ∂  (φ ρ E ) + ∇ · ((φ ρ E + φ p ) u ) = p ∂ φ − λ u · (u − u ) ,  2 2 2 2 2 2 2 2 t 1 2 I I 2 1 ∂t       ∂ φ + u ∇φ = ν(p − p ). 1 1 2 I ∂t 1 (φ1 ρ1 ) + ∇ · (φ1 ρ1 u1 ) = 0,

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

15 / 19

3. Validation

The Baer–Nunziato Model

The Baer-Nunziato Model of Compressible Multi-Phase Flows Definitions k φk uk subscripts 1 subscripts 2 subscripts I ek Ek = ek + 21 uk 2

→ → → → → → → →

phase volume fraction of phase k velocity vector of phase k solid phase gas phase interface internal energy specific total energy of phase k

Assumptions φ1 + φ2 = 1 uI = u1 pI = p2

volume fractions must sum to the unity interface velocity is the solid phase velocity interface pressure is the gas phase pressure a

a

according to the original formulation of the model (M.R. Baer and J.W. Nunziato, J. of Multiphase Flows 1986) Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

15 / 19

3. Validation

The Baer–Nunziato Model

Riemann problems Initial condition for the left state (L) and the right state (R) for the Riemann problems solved with the Baer-Nunziato model. Values for γk , πk and the final time tf are also given. ρs

RP1: L R RP2 : L R RP3 : L R RP4 : L R Walter Boscheri

us ps ρg ug pg φs tf γs = 1.4, πs = 0, γg = 1.4, πg = 0, λ = µ = 0 1.0 0.0 1.0 0.5 0.0 1.0 0.4 0.10 2.0 0.0 2.0 1.5 0.0 2.0 0.8 γs = 3.0, πs = 100, γg = 1.4, πg = 0, λ = µ = 0 800.0 0.0 500.0 1.5 0.0 2.0 0.4 0.10 1000.0 0.0 600.0 1.0 0.0 1.0 0.3 γs = 1.4, πs = 0, γg = 1.67, πg = 0, λ = 103 , µ = 102 1.0 0.0 1.0 1.0 0.0 1.0 0.99 0.2 0.125 0.0 0.1 0.125 0.0 0.1 0.01 γs = 2.0, πs = 0, γg = 1.4, πg = 0, λ = 103 , µ = 102 1.0 0.0 2.0 1.0 0.0 2.0 0.99 0.2 0.125 0.0 0.1 0.125 0.0 0.1 0.01 High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

16 / 19

3. Validation

The Baer–Nunziato Model

Riemann problems RP1 2.2

Reference solution ALE WENO (O3)

2

1.8

rhos

1.6

1.4

1.2

1

0.8 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x

3rd order accurate density distribution for the Riemann problem RP1.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

16 / 19

3. Validation

The Baer–Nunziato Model

Riemann problems RP2 Reference solution ALE WENO (O3)

1050

1000

rhos

950

900

850

800

750 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x

3rd order accurate density distribution for the Riemann problem RP2.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

16 / 19

3. Validation

The Baer–Nunziato Model

Riemann problems RP3 1.1

Exact solution ALE WENO (O3)

1 0.9 0.8

rho

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x

3rd order accurate density distribution for the Riemann problem RP3.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

16 / 19

3. Validation

The Baer–Nunziato Model

Riemann problems RP4 Exact solution ALE WENO (O3)

1.2 1.1 1 0.9 0.8

rho

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x

3rd order accurate density distribution for the Riemann problem RP4.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

16 / 19

3. Validation

The Baer–Nunziato Model

Explosion problems EP1 2.2

Reference solution ALE WENO (O3)

2.1 2 1.9 1.8 1.7

rhos

1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

3rd order accurate density distribution for the explosion problem EP1. Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

17 / 19

3. Validation

The Baer–Nunziato Model

Explosion problems EP2 1050

Reference solution ALE WENO (O3)

1000

rhos

950

900

850

800

750

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

3rd order accurate density distribution for the explosion problem EP2. Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

17 / 19

4. Conclusions and Outlook

Conclusions and Outlook

Conclusions we showed high order ALE WENO FV schemes on tetrahedral meshes; two different node solvers have been used; application to nonconservative hyperbolic balance laws with stiff source terms.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

18 / 19

4. Conclusions and Outlook

Conclusions and Outlook

Conclusions we showed high order ALE WENO FV schemes on tetrahedral meshes; two different node solvers have been used; application to nonconservative hyperbolic balance laws with stiff source terms. Outlook extension to moving curved elements; extension to high order DG direct ALE schemes; application to free surface flows.

Walter Boscheri

High Order ALE ADER Finite Volume Schemes on Unstructured Meshes

18 / 19

Thank you! [email protected] [1] W. Boscheri and M. Dumbser Arbitrary–Lagrangian–Eulerian One–Step WENO Finite Volume Schemes on Unstructured Triangular Meshes. Communications in Computational Physics, 14: 1174-1206, 2013. [2] M. Dumbser and W. Boscheri High-order unstructured Lagrangian one–step WENO finite volume schemes for non–conservative hyperbolic systems: Applications to compressible multi–phase flows. Computers and Fluids, 86: 405-432, 2013. [3] W. Boscheri and M. Dumbser A Direct Arbitrary-Lagrangian-Eulerian ADER-WENO Finite Volume Scheme on Unstructured Tetrahedral Meshes for Conservative and Nonconservative Hyperbolic Systems in 3D. Journal of Computational Physics, 275: 484-523, 2014.