Ph.D. defense
High-order cell-centered discontinuous Galerkin discretizations for scalar conservation laws and Lagrangian hydrodynamics
CEA CESTA1 and INRIA2 | F. Vilar1 , P.-H. Maire1 , R. Abgrall2 16 NOVEMBER 2012
Content
1
Introduction and preliminary results
2
2D Lagrangian hydrodynamics
3
DG discretization general framework
4
Second-order DG scheme
5
Third-order DG scheme
6
Conclusions and perspectives
1/60
Content
1
Introduction and preliminary results
2
2D Lagrangian hydrodynamics
3
DG discretization general framework
4
Second-order DG scheme
5
Third-order DG scheme
6
Conclusions and perspectives
1/60
Motivations and methodology Motivations Inertial confinement fusion −→ compressible gas dynamics simulation
Complex flows (very intense shock and rarefaction waves, strong variation of the fluid domain, multimaterial flows, high cell aspect ratios) Lagrangian formalism (reference frame moving with the fluid) Very high-order extension of the Finite Volume EUCCLHYD scheme P.-H. M AIRE , R. A BGRALL , J. B REIL AND J. OVADIA, A cell-centered Lagrangian scheme for two-dimensional compressible flow problems. SIAM J. Sci. Comput., 2007.
Progressive methodology 1D scalar conservation laws DG discretization 2D scalar conservation laws on unstructured grids DG discretization 1D system of conservation laws DG discretization 2D gas dynamics equation written in a total Lagrangian formalism, on total unstructured grids DG discretization 1/60
Discontinuous Galerkin (DG) DG schemes Natural extension of Finite Volume method Piecewise polynomial approximation of the solution in the cells High-order scheme to achieve high accuracy
Procedure Local variational formulation Choice of the numerical fluxes (global L2 stability, entropy inequality) Time discretization - TVD multistep Runge-Kutta C.-W. S HU, Discontinuous Galerkin methods: General approach and stability. 2008. Limitation - vertex-based hierarchical slope limiters D. K UZMIN, A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods. J. Comp. Appl. Math., 2009. 2/60
1D Scalar Conservation Laws (SCL) Comparison between DG schemes with limitation solution 3rd order 2nd order 1st order
1
0.8
0.6
0.4
0.2
0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
F IGURE: 1D linear advection of a combination of smooth and discontinuous profiles after 10 periods using 200 cells. 3/60
2D Scalar Conservation Laws (SCL) SCL on an unstructured grid made of 2500 polygonal cells 1
1
0.9
0.6
1
0.9
0.8
0.8
0.8
0.4 0.6
0.7
0.7 0.2 0.4
0.6
0.6 0
0.2
0.5
0.5 −0.2
0
0.4
0.4 −0.2 −0.4
0.3
0.3 −0.4 −0.6
0.2
0
0.2
−0.6 −0.8
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a) Burgers problem.
0.9
1
−1
0.1
−0.8 −1 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0
(b) Buckley-Leverett problem.
F IGURE: Numerical solutions using third-order DG scheme with limitation.
J.-L. G UERMOND, R. PASQUETTI AND B. P OPOV, Entropy viscosity method for non-linear conservation laws. J. Comp. Phys., 2011.
4/60
2D Scalar Conservation Laws (SCL)
Third-order DG scheme without limitation L1 Nc 10 × 10 20 × 20 40 × 40 80 × 80 160 × 160
EL1 1.96E-3 2.22E-4 2.75E-5 3.43E-6 4.29E-7
L2 qL1 3.14 3.01 3.00 3.00 -
EL2 2.55E-3 3.00E-4 3.73E-5 4.67E-6 5.83E-7
L∞ qL2 3.09 3.01 3.00 3.00 -
EL∞ 8.07E-3 1.08E-3 1.33E-4 1.65E-5 2.05E-6
qL∞ 2.90 3.02 3.01 3.01 -
TABLE: Rate of convergence in the case of the linear advection (A = (1, 1)t ) of the smooth initial condition u 0 (x) = sin(2πx) sin(2πy ) where x = (x, y )t ∈ [0, 1]2 , with periodic boundary conditions, at the end of a period on Cartesian grids with a CFL= 0.1.
5/60
1D Lagrangian gas dynamics Third-order DG scheme without limitation 0.29
1.1
solution 3rd order
solution 3rd order
0.285
1
0.9
0.28
0.8
0.275
0.7
0.27
0.6
0.265
0.5
0.26
0.4
0.255
0.3
0.25
0.2
0.245
0.24
0.1 0
0.1
0.2
0.3
0.4
0.5
0.6
(a) Global view.
0.7
0.8
0.9
1
0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.84
0.86
(b) Zoom on [0.67, 0.87] × [0.24, 0.29].
F IGURE: Third-order DG scheme solutions for the Sod shock tube problem on 100 cells: density. 6/60
1D Lagrangian gas dynamics Influence of the limitation on the linearized Riemann invariants 0.29
0.29 solution 3rd order
solution 3rd order
0.285
0.285
0.28
0.28
0.275
0.275
0.27
0.27
0.265
0.265
0.26
0.26
0.255
0.255
0.25
0.25
0.245
0.245
0.24
0.24 0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
(a) Physical variables limitation.
0.84
0.86
0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.84
0.86
(b) Riemann invariants limitation.
F IGURE: Third-order DG scheme solutions for the Sod shock tube problem, using 100 cells: density, zoom on [0.67, 0.87] × [0.24, 0.29].
B. C OCKBURN AND C.-W. S HU, The RKDG method for conservation laws V: Multidimensional systems. J. Comp. Phys., 1998. 7/60
1D Lagrangian gas dynamics 3rd order DG scheme with limitation 5 solution 3rd order 4.5
4
3.5
3
2.5
2
1.5
1
0.5 -5
-4
-3
-2
-1
0
1
2
3
4
5
F IGURE: Third-order DG scheme solution with limitation, for a Shu oscillating shock tube problem using 200 cells. 8/60
1D Lagrangian gas dynamics Rate of convergence for the third-order DG scheme
∆X 1 50 1 100 1 200 1 400 1 800
L1 EL1 9.09E-5 1.13E-5 9.40E-7 9.57E-8 1.17E-8
qL1 3.01 3.58 3.30 3.03 -
L2 EL2 3.40E-4 4.64E-5 4.79E-6 4.74E-7 5.63E-8
qL2 2.87 3.28 3.34 3.07 -
L∞ EL∞ 2.20E-3 3.17E-4 4.89E-5 7.85E-6 1.04E-6
qL∞ 2.79 2.70 2.64 2.91 -
TABLE: Rate of convergence computed with the particular smooth solution designed in the special case of γ = 3, on the [0, 1] domain, at time t = 0.8 with a CFL= 0.1.
F. V ILAR , P.-H. M AIRE AND R. A BGRALL, Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics. Comp. & Fluids, 2010. 9/60
Content
1
Introduction and preliminary results
2
2D Lagrangian hydrodynamics
3
DG discretization general framework
4
Second-order DG scheme
5
Third-order DG scheme
6
Conclusions and perspectives
9/60
Cell-Centered Lagrangian schemes Finite volume schemes on moving mesh J. K. Dukowicz: CAVEAT scheme A computer code for fluid dynamics problems with large distorsion and internal slip, 1986
´ GLACE scheme B. Despres: Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems, 2005
P.-H. Maire: EUCCLHYD scheme A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, 2007
G. Kluth: Hyperelasticity Discretization of hyperelasticity with a cell-centered Lagrangian scheme, 2010
S. Del Pino: Curvilinear Finite Volume method A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates, 2010
P. Hoch: Finite Volume method on unstructured conical meshes Extension of ALE methodology to unstructured conical meshes, 2011
DG scheme on initial mesh ` R. Loubere: DG scheme for Lagrangian hydrodynamics A Lagrangian Discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems, 2004 10/60
Lagrangian and Eulerian descriptions Flow transformation of the fluid The fluid flow is described mathematically by the continuous transformation, Φ, so-called mapping such as Φ : X −→ x = Φ(X , t) n
N Φ x = Φ(X, t)
01
X
0011
Ω
∂ω
∂Ω
ω
F IGURE: Notation for the flow map.
where X is the Lagrangian (initial) coordinate, x the Eulerian (actual) coordinate, N the Lagrangian normal and n the Eulerian normal
Deformation Jacobian matrix: deformation gradient tensor F = ∇X Φ =
∂x ∂X
and
J = det F > 0
11/60
Lagrangian and Eulerian descriptions Trajectory equation dx = U(x, t), dt
x(X , 0) = X
Material time derivative d ∂ f (x, t) = f (x, t) + U ∇x f (x, t) dt ∂t
Transformation formulas FdX = dx 0
Change of shape of infinitesimal vectors
ρ = ρJ
Mass conservation
JdV = dv
Measure of the volume change
JF−t NdS = nds
Nanson formula
Differential operators transformations ∇x P = J1 ∇X (P JF−t )
∇x U = J1 ∇X (JF−1 U)
Gradient operator Divergence operator 12/60
Lagrangian and Eulerian descriptions Piola compatibility condition ∇x G = 0, where G = JF−t is the cofactor matrix of F Z Z Z ∇x G dV = G N dS = n ds = 0 Ω
∂Ω
∂ω
Gas dynamics system written in its total Lagrangian form dF − ∇X U = 0 dt d 1 ρ0 ( ) − ∇X (Gt U) = 0 dt ρ dU ρ0 + ∇X (P G) = 0 dt dE + ∇X (Gt PU) = 0 ρ0 dt
Deformation gradient tensor equation Specific volume equation Momentum equation Total energy equation
Thermodynamical closure EOS:
P = P(ρ, ε) where
ε = E − 12 U 2
13/60
Content
1
Introduction and preliminary results
2
2D Lagrangian hydrodynamics
3
DG discretization general framework
4
Second-order DG scheme
5
Third-order DG scheme
6
Conclusions and perspectives
13/60
DG discretization general framework (α + 1)th order DG discretization Let {Ωc }c be a partition of the domain Ω into polygonal cells {σkc }k=0...K basis of P α (Ωc ), where K + 1 = φch (X , t) =
K X
(α+1)(α+2) 2
φck (t)σkc (X ) approximate function of φ(X , t) on Ωc
k=0
Definitions Z 1 ρ0 (X ) X dV , mc Ωc where mc is the constant mass of the cell Ωc Z 1 The mean value hφic = ρ0 (X ) φ(X ) dV mc Ω c of the function φ over the cell Ωc Z The associated scalar product hφ, ψic = ρ0 (X ) φ(X ) ψ(X ) dV Center of mass X c = (Xc , Yc )t =
Ωc
14/60
Polynomial Taylor basis Taylor expansion on the cell, located at the center of mass φ(X ) = φ(X c ) +
q α X q−j j X (X − Xc ) (Y − Yc ) ∂q φ (X c ) + o(kX − X c kα ) j!(q − j)! ∂X q−j ∂Y j q=1 j=0
(α + 1)th order Polynomial Taylor basis The first-order polynomial component and the associated basis function φc0 = hφic
and
σ0c = 1
The q th -order polynomial components and the associated basis functions ∂q φ φcq(q+1) +j = (∆Xc )q−j (∆Yc )j (X c ), ∂X q−j ∂Y j 2 q−j j q−j j X −Xc Y −Yc X −Xc Y −Yc 1 σ cq(q+1) = j!(q−j)! − , ∆Xc ∆Yc ∆Xc ∆Yc 2
+j
where 0 < q ≤ α, j = 0 . . . q, ∆Xc =
c
Xmax −Xmin 2
and ∆Yc =
Ymax −Ymin 2
¨ H. L UO, J. D. B AUM AND R. L OHNER , A DG method based on a Taylor basis for the compressible flows on arbitrary grids. J. Comp. Phys., 2008. 15/60
Polynomial Taylor basis Outcome First moment associated to the basis function σ0c = 1 is the mass averaged value φc0 = hφic The successive moments can be identified as the successive derivatives of the function expressed at the center of mass of the cell φcq(q+1) +j = (∆Xc )q−j (∆Yc )j 2
∂q φ (X c ) ∂X q−j ∂Y j
The first basis function is orthogonal to the other ones hσ0c , σkc ic = mc δ0k Same basis functions regardless the shape of the cells (squares, triangles, generic polygonal cells) 16/60
DG discretization general framework Lagrangian gas dynamics equation type ρ0
dφ + ∇X (Gt f ) = 0, where f is the flux function dt G = JF−t is the cofactor matrix of F
Local variational formulations Z Ωc
ρ0
dφ c σ dV dt q
Z K X d φc
ρ0 σqc σkc dV Ωc k=0 Z Z = f G ∇X σqc dV − =
k
dt
Ωc
f σqc GNdS
∂Ωc
Geometric Conservation Law (GCL) Equation on the first moment of the specific volume Z Z Z dJ d |ωc | t dV = = ∇X (G U) dV = U GNdS dt Ωc dt Ωc ∂Ωc
17/60
DG discretization general framework Mass matrix properties Z Ωc
ρ0 σqc σkc dV = σqc , σkc c
generic coefficient of the symmetric positive
definite mass matrix
c c σ0 , σk c = mc δ0k mass averaged equation is independent of the other polynomial basis components equations
Interior terms Z
f G ∇X σqc dV
is evaluated through the use of a two-dimensional
Ωc
high-order quadrature rule
Boundary terms Z
f σqc GNdS
required a specific treatment to ensure the GCL
∂Ωc
It remains to determine the numerical fluxes 18/60
Entropic analysis Entropic semi-discrete equation Fundamental assumption P U = P U The use of variational formulations and Piola condition leads to Z Z 0 dη ρ θ dV = (P − Ph )(U h − U) GNdS, dt Ωc ∂Ωc where η is the specific entropy and θ the absolute temperature defined by means of the Gibbs identity
Entropic semi-discrete equation Z A sufficient condition to satisfy
ρ0 θ
Ωc
P − Ph = −Z (U − U h )
dη dV ≥ 0 is dt
GN = −Z (U − U h ) n, kGNk
where Z ≥ 0 has the physical dimension of a density times a velocity
19/60
Riemann invariants limitation Riemann invariants differentials associated to unit direction n dαt = dU t
Being given the directions n and t = e z × n
dα− = d( ρ1 ) − dα+ =
d( ρ1 )
+
1 ρa 1 ρa
dU n dU n
dαE = dE − U dU + P d( ρ1 ) a denotes the sound speed
Linearization around the mean values in cell Ωc c αt,h = U ch t c 1 Zc U h n ( ρ1 )ch + Z1c U ch n Ehc − U c0 U ch + P0c
c α−,h = ( ρ1 )ch − c α+,h = c αE,h =
( ρ1 )ch
where Zc = a0c ρc0 is the acoustic impedance
20/60
Riemann invariants limitation
System variables polynomial approximation components c c ) + α−,k ( ρ1 )ck = 21 (α+,k c c c t )n + αt,k − α−,k U ck = 12 Zc (α+,k c c c c c c Ekc = αE,k + 21 Zc (α+,k − α−,k )U c0 n + αt,k U c0 t − 12 P0c (α+,k + α−,k )
Unit direction ensuring symmetry preservation n=
U c0 kU c0 k
and
t = ez ×
U c0 kU c0 k
21/60
Deformation gradient tensor discretization Requirements Consistency of vector GNdS = nds at the interfaces of the cells Continuity of vector GN at cell interfaces on both sides of the interface Preservation of uniform flows, G = JF−t the cofactor matrix Z Z Z c c G∇X σq dV = σq GNdS ⇐⇒ σqc (∇X G) dV = 0 Ωc
∂Ωc
Ωc
Generalization of the weak form of the Piola compatibility condition
Tensor F discretization Discretization of tensor F by means of a mapping defined on triangular cells Partition of the polygonal cells in the initial configuration into non-overlapping triangles Ωc =
ntri [ i=1
Ti c
Ωc Tic
22/60
Deformation gradient tensor discretization Continuous mapping function We develop Φ on the Finite Elements basis functions λp X Φih (X , t) = λp (X ) Φp (t), p
where the points p are control points including vertices in Ti Φp (t) = Φ(X p , t) = x p
X d Φp d = U p =⇒ Fi (X , t) = U p (t) ⊗ ∇X λp (X ) dt dt p G. K LUTH AND B. D ESPR E´ S, Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. J. Comp. Phys., 2010.
Outcome Satisfaction of the Piola compatibility condition everywhere Consistency and continuity of the Eulerian normal GN
23/60
Content
1
Introduction and preliminary results
2
2D Lagrangian hydrodynamics
3
DG discretization general framework
4
Second-order DG scheme
5
Third-order DG scheme
6
Conclusions and perspectives
23/60
Geometry discretization P1 barycentric coordinate basis functions In a generic triangle Ti λp (X ) =
1 [X (Yp+ − Yp− ) − Y (Xp+ − Xp− ) + Xp+ Yp− − Xp− Yp+ ], 2|Ti |
where p, p+ and p− are the counterclockwise ordered triangle nodes and |Ti | the triangle volume
Deformation gradient tensor discretization Φih (X , t) =
X
p−
λp (X ) x p (t),
p∈P(Ti )
Lp + p − N p + p −
where P(Ti ) is the node set of Ti 1 X d Fi (t) = U p (t) ⊗ Lpi N pi dt |Ti | p∈P(Ti )
Ti p
p+
LpiN pi 24/60
Local variational formulations DG discretization of the Lagrangian gas dynamics equations type Gci = (JF−t )ci is constant on Ti c and ∇X σq constant over Ωc Z Z ntri X X Z p+ 0dφ c c c f σqc GNdL ρ σ dV = − Gi ∇ X σ q f dV + dt q Ωc Ti c p i=1
Linear assumptions on face fpp+ c f |pp+ (ζ)
− = f+ pc (1 − ζ) + f p+ c ζ,
f+ pc
p∈P(c)
Initial configuration cell p+
f− p+ c
where and are respectively the right and left nodal numerical fluxes
Linear property on face fpp+ c σq|
pp+
N pp+ f− p+ c
Lpp+
p
f+ pc Ωc
f− pc Lp− p
N p− p
(ζ) = σqc (X p ) (1 − ζ) + σqc (X p+ ) ζ,
where σqc (X p ) and σqc (X p+ ) are the extrapolated values of the function σqc
p− 25/60
DG discretization Fundamental assumptions U± pc = U p , PU = P U
∀c ∈ C(p) =⇒
± (PU)± pc = Ppc U p
Procedure Analytical integration + index permutation
Weighted corner normals q q −,q −,q +,q +,q lpc npc = lpc npc + lpc npc 1 +,q +,q 2σqc (X p ) + σqc (X p+ ) lpp+ npp+ lpc npc = 6 1 −,q −,q npc = lpc 2σqc (X p ) + σqc (X p− ) lp− p np− p 6
q th moment of the subcell forces − −,q −,q + +,q +,q lpc npc + Ppc lpc npc F qpc = Ppc
26/60
Local variational formulations Semi-discrete equations GCL compatible Z
ntri
X d 1 c ( )σq dV = − Gci ∇X σqc ρ dt ρ Ωc 0
i=1
Z
ρ0
Ωc
Z Ωc
ρ0
dU c σ dV = dt q dE c σ dV = dt q
ntri X
Gci ∇X σqc
Z
i=1
Ti c
ntri X
Z
Gci ∇X σqc
i=1
Z UdV + Ti c
PdV −
Ti
c
X
q q U p lpc npc
p∈P(c)
X
F qpc
p∈P(c)
P UdV −
X
U p F qpc
p∈P(c)
First moment equations X d 1 c 0 0 ( )0 = U p lpc npc dt ρ p∈P(c) X d E0c mc =− U p F 0pc dt mc
mc
X 0 d U c0 =− F pc dt p∈P(c)
We recover the EUCCLHYD scheme
p∈P(c)
27/60
Nodal solvers q th moment of the subcell forces The use of P = Phc − Zc (U − U ch ) n
to calculate F qpc leads to
q q F qpc = Phc (X p , t) lpc npc − Mqpc (U p − U ch (X p , t)),
where Mqpc = Zc
−,q −,q +,q +,q +,0 lpc npc ⊗ n−,0 pc + lpc npc ⊗ npc
Momentum and total energy conservation X
F 0pc = 0
c∈C(p)
Nodal velocity (
X
c∈C(p)
0 Mpc ) Up =
X 0 0 Phc (X p , t) lpc npc + M0pc U ch (X p , t) c∈C(p) 28/60
Numerical results Sedov point blast problem on a Cartesian grid 5.5
solution 2nd order
6
5
1
4.5
5
4
0.8
4 3.5 3
0.6
3 2.5 2
0.4
2
1.5 1
0.2
1
0.5 0
0 0
0.2
0.4
0.6
0.8
1
(a) Second-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
F IGURE: Point blast Sedov problem on a Cartesian grid made of 30 × 30 cells: density. 29/60
Numerical results Sedov point blast problem on unstructured grids
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
0.8
(a) Polygonal grid.
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
(b) Triangular grid.
F IGURE: Unstructured initial grids for the point blast Sedov problem.
30/60
Numerical results Sedov point blast problem a polygonal grid solution 2nd order
5
6 4.5 1 4
5
3.5
0.8
4 3 0.6
2.5
3
2 0.4
2 1.5 1
0.2
1
0.5 0
0 0
0.2
0.4
0.6
0.8
1
(a) Second-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
F IGURE: Point blast Sedov problem on an unstructured grid made of 775 polygonal cells: density map. 31/60
Numerical results Sedov point blast problem on a triangular grid 5.5
solution 2nd order
6
5
1
4.5
5
4
0.8
4 3.5 3
0.6
3 2.5 0.4
2
2
1.5 0.2
1
1
0.5 0
0 0
0.2
0.4
0.6
0.8
1
(a) Second-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
F IGURE: Point blast Sedov problem on an unstructured grid made of 1100 triangular cells: density map. 32/60
Numerical results Noh problem 16
exact solution 2nd order
16 0.5
14
14
12
12
10
10
0.4
0.3
8 8
6
0.2 6
4 4
0.1
2 2 0
0 0
0.1
0.2
0.3
0.4
0.5
(a) Second-order scheme.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(b) Density profile.
F IGURE: Noh problem on a Cartesian grid made of 50 × 50 cells: density. 33/60
Numerical results Taylor-Green vortex problem, introduced by R. Rieben (LLNL) 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 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a) Second-order scheme.
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Exact solution.
F IGURE: Motion of a 10 × 10 Cartesian mesh through a T.-G. vortex, at t = 0.75.
34/60
Rate of convergence computed on the pressure in the case of the Taylor-Green vortex h 1 20 1 40 1 80 1 160 1 320
L1 ELh1 1.32E-2 3.84E-3 1.01E-3 2.55E-4 6.38E-5
qLh1 1.78 1.93 1.99 2.00 -
L2 ELh2 1.96E-2 6.66E-3 1.80E-3 4.57E-4 1.14E-4
qLh2 1.56 1.89 1.98 2.00 -
L∞ ELh∞ 7.41E-2 3.63E-2 1.21E-2 3.31E-3 8.47E-4
qLh∞ 1.03 1.58 1.87 1.97 -
TABLE: Second-order MUSCL scheme without limitation at time t = 0.6.
h 1 20 1 40 1 80 1 160 1 320
L1 ELh1 8.98E-3 2.44E-3 6.36E-4 1.59E-4 3.94E-5
qLh1 1.88 1.94 2.00 2.01 -
L2 ELh2 1.51E-2 4.48E-3 1.16E-3 2.90E-4 7.18E-5
qLh2 1.75 1.95 2.00 2.01 -
L∞ ELh∞ 6.73E-2 2.79E-2 8.68E-3 2.24E-3 5.54E-4
qLh∞ 1.27 1.68 1.95 2.01 -
TABLE: Second-order DG scheme without limitation at time t = 0.6.
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Content
1
Introduction and preliminary results
2
2D Lagrangian hydrodynamics
3
DG discretization general framework
4
Second-order DG scheme
5
Third-order DG scheme
6
Conclusions and perspectives
35/60
Curvilinear elements motivation Circular polar grid: 10 × 1 cells
Taylor-Green exact motion
1
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1
V. D OBREV, T. E LLIS , T. KOLEV AND R. R IEBEN, High Order Curvilinear Finite Elements for Lagrangian Hydrodynamics. Part I: General Framework, 2010. Presentation available at https://computation.llnl.gov/casc/blast/blast.html 36/60
Geometry discretization P2 Finite Elements basis functions The P2 barycentric coordinate functions µp write µp = (λp )2 , µp+ = (λp+ )2 , µp− = (λp− )2 , µQ = 2λp λp+ , µQ + = 2λp+ λp− , µQ − = 2λp− λp , where {λl }l∈P(Ti ) is the P1 Finite Elements linear basis
Mapping discretization Φ(X , t) =
X
x q (t) µq (X ) =
q
X
x p (t) (λp (X ))2 + 2x Q (t) λp (X )λ+ p (X )
p∈P(Ti )
Deformation gradient tensor discretization d 2 X Fi (X , t) = λp (X ) U p ⊗ Lpc N pc + U Q ⊗ Lp+ c N p+ c + U − Q ⊗ Lp− c N p− c dt |Ti | p∈P(Ti )
37/60
Geometric consideration Mapping of the fluid flow: transformation of Ti into τi p+
p+
Φ
Q
+
Q+
m+
Q
Ti p
−
Q−
p−
p
τi
Q Q− m m− p
Bezier curves Given the three points p, Q and p+ , and ζ ∈ [0, 1] x(ζ) = (1 − ζ)2 x p + 2ζ(1 − ζ)x Q + ζ 2 x p+
= (1 − ζ)(1 − 2ζ)x p + 4ζ(1 − ζ)x m + ζ(2ζ − 1)x p+
Midpoint
x m = x( 12 ) =
Tangent
tdl =
2x Q + x p + x p+ 4
dx dζ = 2 ((1 − ζ)(x Q − x p ) + ζ(x p+ − x Q )) dζ dζ
38/60
DG discretization Local variational formulations ntri
Z
ρ0
Ωc
X dφ c σq dV = − dt +
Ti c
X Z
p∈P(c)
N pp+ f− p+ c
Z
i=1
p+
G ∇X σqc f dV
f mc Ωc
p+
f
σqc
p
f+ pc f− pc
N p− p
GNdL
p p−
Quadratic assumptions on face fpp+ − f |pp+ (ζ) = (1 − ζ)(1 − 2ζ) f + pc + 4ζ(1 − ζ) f mc + ζ(2ζ − 1) f p+ c
Linear and quadratic properties on face fpp+ G N dL|pp+ (ζ) = 2 (1 − ζ) lpQ npQ + ζ lQp+ nQp+ dζ c σq|
pp+
(ζ) = (1 − ζ)(1 − 2ζ) σqc (X p ) + 4ζ(1 − ζ) σqc (X m ) + ζ(2ζ − 1) σqc (X p+ ) 39/60
DG discretization Semi-discrete equations GCL compatible Z
ρ0
Ωc
Z
i=1
ρ0
Ωc
Z
ntri Z X X1 X 2 d 1 c q q q ( )σq dV = − U G∇X σqc dV + U p lpc U m lmc npc + nqmc c dt ρ 3 3 Ti
ρ0
Ωc
dU c σ dV = dt q
p∈P(c)
ntri Z X i=1
Ti
ntri Z X
dE c σ dV = − dt q
i=1
c
PG∇X σqc dV −
X 1 q F − 3 pc
p∈P(c)
PU G∇X σqc dV +
Ti c
m∈M(c)
X m∈M(c)
2 q F 3 mc
X 1 X U p F qpc + 3
p∈P(c)
m∈M(c)
2 U m F qmc 3
Equation on the first moment of the specific volume d |ωc | = dt
Z U GNdL = ∂Ωc
X 1 U p lQ − Q n Q − Q + 3
p∈P(c)
X m∈M(c)
2 U m lpp+ npp+ 3
B. B OUTIN , E. D ERIAZ , P. H OCH , P. N AVARO, Extension of ALE methodology to unstructured conical meshes, ESAIM: Proceedings, 2011 40/60
Nodal and midpoint solvers Subcell forces definitions − −,q −,q + +,q +,q F qpc = Ppc lpc npc + Ppc lpc npc
q and F qmc = Pmc lmc nqpc
q th moment of the nodal and midpoint subcell forces The use of P = Phc − Zc (U − U ch ) n
to calculate F qpc and F qmc leads to
q q F qpc = Phc (X p , t) lpc npc − Mqpc (U p − U ch (X p , t)), q F qmc = Phc (X m , t) lmc nqmc − Mqmc (U m − U ch (X m , t)),
Mqpc = Zc
−,q −,q +,q +,q q −,0 lpc npc ⊗ npc + lpc npc ⊗ n+,0 and Mqmc = Zc lmc nqmc ⊗ n0mc pc
Momentum and total energy conservation X
F 0pc = 0
and
F 0mL + F 0mR = 0
c∈C(p) 41/60
Nodal and midpoint solvers Nodal velocity Mp U p =
X 0 0 Phc (X p , t) lpc npc + M0pc U ch (X p , t) , c∈C(p)
where
X
Mp =
M0pc
is a positive definite matrix
c∈C(p)
Midpoint velocity Mm U m = Mm where Mm =
ZL U Lh (X m ) + ZR U R h (X m ) ZL + ZR
1 0 ZL MmL
=
1 0 ZR MmR
! −
PhR (X m ) − PhL (X m ) 0 0 lmc nmc , ZL + ZR
0 n0mc ⊗ n0mc is positive semi-definite = lmc
1D approximate Riemann problem solution (U m
n0mc )
=
ZL U Lh (X m ) + ZR U R h (X m ) ZL + ZR
! n0mc −
PhR (X m ) − PhL (X m ) ZL + ZR 42/60
Nodal and midpoint solvers Tangential component of the midpoint velocity (U m
t 0mc )
=
ZL U Lh (X m ) + ZR U R h (X m ) ZL + ZR
! t 0mc
Midpoint velocity Um =
P R (X m ) − PhL (X m ) 0 ZL U Lh (X m ) + ZR U R h (X m ) − h nmc ZL + ZR ZL + ZR
Control point velocity UQ =
4U m − U p − U p+ 2
1 0 0 1 0 1 1 0 0 1
0 1 1 0 0 1
11 00 11 00
Φ 1 0
11 00
Interior points velocity
1 0 1 0
0 1 1 0 0 1
11 00 00 11 00 11
00 11 11 00 00 11
1 0 1 0 11 00
Ωc 11 00 00 11
11 00
1 0 0 1 0 1
11 00 00 11 11 00
U i = U ch (X i , t)
1 0 0 1
1 0 1 0 11 00 00 11 00 11
1 0 0 1 0 1 11 00 00 11
11 00 11 00
1 0 0 1 0 1
ωc 11 00 11 00
1 0 1 0
43/60
Deformed initial mesh Composed derivatives Ωc
FT = ∇Xr ΦT (X r , t)
X
= ∇X ΦH (X , t) ◦ ∇Xr Φ0 (X r )
= FH F0
JT (X r , t) = JH (X , t) J0 (X r )
ΦH (X, t)
Ωrc
Φ0(X r )
Xr x
Mass conservation ρ0 J0 = ρ JT
ωc ΦT (X r , t)
Modification of the mass matrix Z K X d ψhc d ψk σq dω = ρ0 J0 σq σk dΩr r dt dt ωc Ωc k =0 successive moments of function ψ Z
ρ
time rate of change of
New definitions of mass matrix, of mass averaged value and of the associated scalar product 44/60
Numerical results One angular cell polar Sod shock tube problem 1
1
1.1 solution 3rd order
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(a) Density map.
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(b) Density profile.
F IGURE: Third-order DG solution for a Sod shock tube problem on a polar grid made of 100 × 1 cells. 45/60
Numerical results Symmetry preservation 1
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(b) Non-uniform grid.
F IGURE: Polar initial grids for the Sod shock tube problem. 46/60
Numerical results Symmetry preservation 1
1
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(a) First-order scheme.
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(b) Second-order scheme.
F IGURE: Sod shock tube problem on a polar grid made of 100 × 3 cells. 47/60
Numerical results Symmetry preservation 1
1
1.1 solution 3rd order
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1 0.9
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(a) Density map.
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(b) Density profile.
F IGURE: Third-order DG solution for a Sod shock tube problem on a polar grid made of 100 × 3 cells. 48/60
Numerical results Symmetry preservation 1
1
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(a) First-order scheme.
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(b) Second-order scheme.
F IGURE: Sod shock tube problem on a polar grid made of 100 × 3 non-uniform cells. 49/60
Numerical results Symmetry preservation 1
1
1.1 solution 3rd order
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(a) Density map.
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(b) Density profile.
F IGURE: Third-order DG solution for a Sod shock tube problem on a polar grid made of 100 × 3 non-uniform cells. 50/60
Numerical results Variant of the incompressible Gresho vortex problem 0.5
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(a) First-order scheme.
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(b) Second-order scheme.
F IGURE: Motion of a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1: zoom on the zone (r , θ) ∈ [0, 0.5] × [0, 2π]. 51/60
Numerical results Variant of the incompressible Gresho vortex problem 0.5
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(a) Third-order scheme.
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(b) Exact solution.
F IGURE: Motion of a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1: zoom on the zone (r , θ) ∈ [0, 0.5] × [0, 2π]. 52/60
Numerical results Variant of the Gresho vortex problem 5.6
1 solution 1st order 2nd order 3rd order
solution 1st order 2nd order 3rd order
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5.3
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0.2 5.1 0.1
5
0 0
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(a) Pressure profile.
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(b) Velocity profile.
F IGURE: Gresho variant problem on a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1.
53/60
Numerical results Variant of the Gresho vortex problem 1.06 solution 1st order 2nd order 3rd order
1.05
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1
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1
F IGURE: Gresho variant problem on a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1: density profile. 54/60
Numerical results Kidder isentropic compression 1
250
9
0.4 0.9 8
0.35
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200
7 0.7
0.3 6
0.6
150
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0.2 4 0.4
100 0.15
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3
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1
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0 0
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(a) At time t = 0.
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(b) At time t = 0.9τ .
F IGURE: Third-order DG solution for a Kidder isentropic compression problem on a polar grid made of 10 × 3 cells: pressure map. 55/60
Numerical results Kidder isentropic compression 0.055 solution 3rd order 0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
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0.005 0.39
0.395
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0.405
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0.415
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0.425
0.43
0.435
F IGURE: Third-order DG solution for a Kidder isentropic compression problem on a polar grid made of 10 × 3 cells: density profile. 56/60
Numerical results Taylor-Green vortex problem 1
1
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(a) Third-order scheme.
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(b) Exact solution.
F IGURE: Motion of a 10 × 10 Cartesian mesh through a T.-G. vortex, at t = 0.75.
57/60
Rate of convergence computed on the pressure in the case of the Taylor-Green vortex h 1 10 1 20 1 40 1 80 1 160
L1 ELh1 2.50E-2 8.98E-3 2.44E-3 6.36E-4 1.59E-4
qLh1 1.48 1.88 1.94 2.00 2.01
L2 ELh2 3.71E-2 1.51E-2 4.48E-3 1.16E-3 2.90E-4
qLh2 1.30 1.75 1.95 2.00 2.01
L∞ ELh∞ 1.72E-1 6.73E-2 2.79E-2 8.68E-3 2.24E-3
qLh∞ 1.35 1.27 1.68 1.95 2.01
TABLE: Second-order DG scheme without limitation at time t = 0.6.
h 1 10 1 20 1 40 1 80 1 160
L1 ELh1 4.39E-3 5.50E-4 6.68E-5 8.90E-6 1.20E-6
qLh1 3.00 3.04 2.91 2.89 -
L2 ELh2 7.73E-3 1.21E-3 1.40E-4 1.92E-5 2.70E-6
qLh2 2.68 3.10 2.87 2.83 -
L∞ ELh∞ 3.90E-2 1.03E-2 1.30E-3 2.11E-4 3.16E-5
qLh∞ 1.93 2.98 2.66 2.74 -
TABLE: Third-order DG scheme without limitation at time t = 0.6.
58/60
Rate of convergence computed on the pressure in the case of the Taylor-Green vortex
L1 h 1 10 1 20 1 40 1 80 1 160
L2
L∞
ELh1
qLh1
ELh2
qLh2
ELh∞
2.67E-4 3.43E-5 4.37E-6 5.50E-7 6.91E-8
2.96 2.97 2.99 2.99 -
3.36E-4 4.36E-5 5.59E-6 7.06E-7 8.87E-8
2.94 2.96 2.98 2.99 -
1.21E-3 1.66E-4 2.18E-5 2.80E-6 3.53E-7
qLh∞ 2.86 2.93 2.96 2.99 -
TABLE: Third-order DG scheme without limitation at time t = 0.1.
59/60
Content
1
Introduction and preliminary results
2
2D Lagrangian hydrodynamics
3
DG discretization general framework
4
Second-order DG scheme
5
Third-order DG scheme
6
Conclusions and perspectives
59/60
Conclusions and perspectives Conclusions DG schemes up to 3rd order 1D and 2D scalar conservation laws on general unstructured grids 1D systems of conservation laws 2D gas dynamics system in a total Lagrangian formalism
GCL and Piola compatibility condition ensured by construction Dramatic improvement of symmetry preservation by means of third-order DG scheme
Perspectives High-order limitation on curved geometries Improvement in midpont solver definition Code parallelization Study on computational cost and time Development of a 3rd order DG scheme on moving mesh Extension to 3D Extension to ALE and solid dynamics
60/60
Articles
F. V ILAR , P.-H. M AIRE AND R. A BGRALL, Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics. Computers and Fluids, 2010. F. V ILAR, Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics. Computers and Fluids, 2012. F. V ILAR , P.-H. M AIRE AND R. A BGRALL, Third order Cell-Centered DG scheme for Lagrangian hydrodynamics on general unstructured Bezier grids. Article in preparation.
60/60
Thank you 0.5
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Third-order polynomial Taylor basis Taylor expansion on the cell, located at the center of mass X c ∂φ ∂φ 1 ∂2 φ + (Y − Yc ) + (X − Xc )2 ∂X ∂Y 2 ∂X 2 2 1 ∂ φ ∂2 φ + (Y − Yc )2 +(X − Xc )(Y − Yc ) + o(kX − X c k2 ) ∂X ∂Y 2 ∂Y 2
φ(X ) = φ(X c ) + (X − Xc )
Polynomial basis functions σ0c = 1, σ1c =
X −Xc ∆Xc ,
σ2c =
Y −Yc ∆Yc ,
σ3c =
1 2
X −Xc ∆Xc
2
−
X −Xc ∆Xc
2 , c E
D c )(Y −Yc ) − (X −X , ∆Xc ∆Yc c 2 2 Y −Yc Y −Yc σ5c = 12 − . ∆Yc ∆Yc σ4c =
(X −Xc )(Y −Yc ) ∆Xc ∆Yc
c
Polynomial apprixation function components ∂φ ∂φ c c ∂X (X c ), φ2 = ∆Yc ∂Y (X c ), φ3 2 2 ∂ φ c 2 ∂ φ ∆Xc ∆Yc ∂X ∂Y (X c ), φ5 = (∆Yc ) ∂Y 2 (X c )
φc0 = hφic , φc1 = ∆Xc φc4 =
= (∆Xc )2
∂2 φ (X c ), ∂X 2 60/60
Vertex-based hierarchical slope limiter Third-order DG scheme limitation φch = φc0 + c1 (φc1 σ1c + φc2 σ2c ) + c2 (φc3 σ3c + φc4 σ4c + φc5 σ5c ) where c1 and c2 are the limiting coefficients
Linear reconstruction φ(1) = φc0 + c1 φc1 (2)
∂ φch ∂X
(2)
∂ φch ∂Y
φX = ∆Xc φY = ∆Yc
−Yc + φc2 Y∆Y c −Xc = φc1 + cX φc3 X∆X + φc4 c −Xc = φc2 + cY φc4 X∆X + φc5 c X −Xc ∆Xc
Y −Yc ∆Yc
Y −Yc ∆Yc
Limiting coefficient c2 = min(cX , cY ) c1 = max(c1 , c2 )
Smooth extrema preservation 60/60
Riemann invariants limitation Riemann invariants differentials dαt = dU t, dα− = dP − ρa dU n, dα+ = dP + ρa dU n,
dαE = dE − U dU + P d( ρ1 ), where n denotes a unit vector and t = e z × n
Isentropic flow dP = −ρ2 a2 d( ρ1 )
New Riemann invariants differentials dαt = dU t, dα− = d( ρ1 ) − dα+ =
d( ρ1 )
+
1 ρa 1 ρa
dU n, dU n,
dαE = dE − U dU + P d( ρ1 )
60/60
2nd order DG discretization DG discretization of the Lagrangian gas dynamics equations type Gci = (JF−t )ci is constant on Ti c and ∇X σq over Ωc Z Z ntri X X Z p+ 0dφ c c c f σqc GNdL ρ σ dV = − Gi ∇ X σ q f dV + dt q Ωc Ti c p i=1
Z
ρ0
Ωc
dψ c σ dV = − dt q
p∈P(c)
ntri X
Gci ∇X σqc
i=1
Z h dV + Ti c
Linear assumptions on face fpp+ c f |pp+ (ζ) c h|pp+ (ζ)
=
(1 − ζ) +
hp−+ c
pp+
p+
(ζ) = σqc (X p ) (1 − ζ) + σqc (X p+ ) ζ
h σqc GNdL
p
N pp+ f− p+ c
Lpp+
p
f+ pc
ζ
Linear property on face fpp+ c σq|
p∈P(c)
− = f+ pc (1 − ζ) + f p+ c ζ + hpc
p+
X Z
Ωc
f− pc Lp− p
N p− p
p−
60/60
2nd order DG discretization Analytical integration Z p
p+
f σqc GNdL =
Z 0
!
1 c f |pp+ (ζ) σq| (ζ) dζ pp+
G|pp+ Lpp+ N pp+
G|pp+ Lpp+ N pp+ = lpp+ npp+ Eulerian normal of face fpp+ Z X 1 + f σqc GNdL = f pc 2σqc (X p ) + σqc (X p+ ) lpp+ npp+ 6 ∂Ωc p∈P(c)
c c +f − p+ c (2σq (X p+ ) + σq (X p ))lpp+ npp+
Weighted corner normals q q −,q −,q +,q +,q lpc npc = lpc npc + lpc npc 1 +,q +,q lpc npc = 2σqc (X p ) + σqc (X p+ ) lpp+ npp+ 6 1 −,q −,q lpc npc = 2σqc (X p ) + σqc (X p− ) lp− p np− p 6 60/60
2nd order DG discretization Index permutation Z
X
f σqc GNdL =
∂Ωc
Z
−,q −,q +,q +,q + f− pc lpc npc + f pc lpc npc
p∈P(c)
h σqc
GNdL =
∂Ωc
X − −,q −,q + +,q +,q hpc lpc npc + hpc lpc npc p∈P(c)
Numerical fluxes on face fpp+ c
U |pp+ (ζ) = U p (1 − ζ) + U p+ ζ c
c
+ P |pp+ (ζ) = Ppc (1 − ζ) + Pp−+ c ζ
− PU |pp+ (ζ) = (PU)+ pc (1 − ζ) + (PU)p+ c ζ
Fundamental assumption on face fpp+ PU =PU
=⇒
− + + (PU)− pc = Ppc U p and (PU)pc = Ppc U p
q th moment of the subcell forces − −,q −,q + +,q +,q F qpc = Ppc lpc npc + Ppc lpc npc
60/60
2nd order DG discretization Semi-discrete equations GCL compatible Z
ntri
X d 1 c ( )σq dV = − Gci ∇X σqc ρ dt ρ Ωc 0
i=1
Z
ρ0
Ωc
Z Ωc
ρ0
dU c σ dV = dt q dE c σ dV = dt q
ntri X
Gci ∇X σqc
Z
i=1
Ti c
ntri X
Z
Gci ∇X σqc
i=1
Z UdV + Ti c
PdV −
Ti
c
X
q q U p lpc npc
p∈P(c)
X
F qpc
p∈P(c)
P UdV −
X
U p F qpc
p∈P(c)
First moment equations X d 1 c ( )0 = U p lpc npc dt ρ p∈P(c) X d E0c mc =− U p F pc dt mc
mc
X d U c0 =− F pc dt p∈P(c)
We recover the EUCCLHYD scheme
p∈P(c)
60/60
Compatibility of deformation gradient tensor discretization Theoretical compatibility dF = ∇X U dt ∂ dF dF dF dJ = (det F) : = (det F)F−t : = JF−t : dt ∂F dt dt dt dJ = JF−t : ∇X U = JF−t : (∇x U) (∇X x) = JF−t Ft : ∇x U dt = J tr(∇x U) = J∇x U = ∇X (JF−1 U) = ∇X (Gt U) dJ d ρ0 d 1 = = ρ0 = ∇X (Gt U) dt dt ρ dt ρ
Second-order discretizations compatibility X X d Jic d Fci 1 1 = Gci : = c U p Gci Lpi N pi = c U p lpi npi dt dt |Ti | |Ti | c p∈P(Ti ) p∈P(Ti ) c c ntri ntri c X d Ji |τi | d |ωc | 1 1 1 X c c c = thus = = |τ |Ti | Ji | = i dt dt |Ti c | ρ 0 mc mc mc i=1
i=1
60/60
3rd order DG discretization DG discretization of the Lagrangian gas dynamics equations type Z ρ
0dφ
dt
Ωc
Z
ρ0
Ωc
σqc
dV = −
ntri Z X
dψ c σ dV = − dt q
i=1
Ti c
ntri Z X i=1
Ti c
G ∇X σqc
f dV +
p∈P(c)
G ∇X σqc h dV +
p+
X Z
f σqc GNdL
p
X Z p∈P(c)
p+
h σqc GNdL
p
Quadratic assumptions on face fpp+ − f |pp+ (ζ) = (1 − ζ)(1 − 2ζ) f + pc + 4ζ(1 − ζ) f mc + ζ(2ζ − 1) f p+ c + h|pp+ (ζ) = (1 − ζ)(1 − 2ζ) hpc + 4ζ(1 − ζ) hmc + ζ(2ζ − 1) hp−+ c
Linear and quadratic properties on face fpp+ G N dL|pp+ (ζ) = n dl|pp+ (ζ) = 2 (1 − ζ) lpQ npQ + ζ lQp+ nQp+ dζ c σq|
pp+
(ζ) = (1 − ζ)(1 − 2ζ) σqc (X p ) + 4ζ(1 − ζ) σqc (X m ) + ζ(2ζ − 1) σqc (X p+ ) 60/60
3rd order DG discretization Analytical integration + Index permutation Z
f σqc GNdL =
∂Ωc
X 2 X 1 − −,q −,q q +,q +,q f pc lpc npc + f + f mc lmc nqmc l n + pc pc pc 3 3
p∈P(c)
m∈M(c)
Z
X 2 X 1 q − −,q −,q + +,q +,q hσqc GNdL = hpc lpc npc + hpc hmc lmc nqmc lpc npc + 3 3 ∂Ωc p∈P(c)
m∈M(c)
Weighted midpoint and corner normals
q −,q −,q +,q +,q q q −,q −,q +,q +,q lmc nqmc = lmc nmc + lmc nmc and lpc npc = lpc npc + lpc npc 1 −,q − lmc nmc = 4 σqc (X m ) + σqc (X p ) lpQ npQ 5 1 +,q + lmc nmc = 4 σqc (X m ) + σqc (X p+ ) lQp+ nQp+ 5 1 −,q −,q (6 σqc (X p ) + 4 σqc (X m− ))lQ − p nQ − p + (σqc (X p ) − σqc (X p− ))lp− p np− p lpc npc = 10 1 +,q +,q lpc npc = (6 σqc (X p ) + 4 σqc (X m ))lpQ npQ + (σqc (X p ) − σqc (X p+ ))lpp+ npp+ 10 60/60
3rd order DG discretization Semi-discrete equations GCL compatible Z
ρ0
Ωc
Z
i=1
ρ0
Ωc
Z
ntri Z X X1 X 2 d 1 c q q q ( )σq dV = − U G∇X σqc dV + U p lpc U m lmc npc + nqmc c dt ρ 3 3 Ti
ρ0
Ωc
dU c σ dV = dt q
p∈P(c)
ntri Z X i=1
Ti
ntri Z X
dE c σ dV = − dt q
i=1
c
PG∇X σqc dV −
X 1 q F − 3 pc
p∈P(c)
PU G∇X σqc dV +
Ti c
m∈M(c)
X m∈M(c)
2 q F 3 mc
X 1 X U p F qpc + 3
p∈P(c)
m∈M(c)
2 U m F qmc 3
Equation on the first moment of the specific volume Z U GNdL = ∂Ωc
X 1 U p lQ − Q nQ − Q + 3
p∈P(c)
X m∈M(c)
2 U m lpp+ npp+ 3
B. B OUTIN , E. D ERIAZ , P. H OCH , P. N AVARO, Extension of ALE methodology to unstructured conical meshes, ESAIM: Proceedings, 2011 60/60
Rate of convergence computed on the pressure in the case of the Taylor-Green vortex D.O.F 600 2400
N 24 × 25 48 × 50
ELh1 2.67E-2 1.36E-2
ELh2 3.31E-2 1.69E-2
ELh∞ 8.55E-2 4.37E-2
time (sec) 2.01 11.0
TABLE: First-order DG scheme at time t = 0.1.
D.O.F 630 2436
N 14 × 15 28 × 29
ELh1 2.76E-3 7.52E-4
ELh2 3.33E-3 9.02E-4
ELh∞ 1.07E-2 2.73E-3
time (sec) 2.77 11.3
TABLE: Second-order DG scheme without limitation at time t = 0.1.
D.O.F 600 2400
N 10 × 10 20 × 20
ELh1 2.67E-4 3.43E-5
ELh2 3.36E-4 4.36E-5
ELh∞ 1.21E-3 1.66E-4
time (sec) 4.00 30.6
TABLE: Third-order DG scheme without limitation at time t = 0.1. 60/60
Numerical results Sedov point blast problem on a Cartesian grid 5.5
solution 3rd order
6 5 1 4.5
5 4 0.8 3.5
4
3 0.6
3 2.5 2 0.4
2 1.5 1
0.2
1
0.5
0
0 0
0.2
0.4
0.6
0.8
1
(a) Third-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
F IGURE: Point blast Sedov problem on a Cartesian grid made of 30 × 30 cells: density. 60/60
