Introduction
2D Lagrangian hydrodynamics
Conclusion
Cell-centered discontinuous Galerkin scheme for Lagrangian hydrodynamics F. Vilar1 , P. H. Maire1 , R. Abgrall2 1 CEA
CESTA, BP 2, 33 114 Le Barp, France
2 INRIA
and University of Bordeaux, Team Bacchus, ´ Institut de Mathematiques de Bordeaux, ´ 351 Cours de la Liberation, 33 405 Talence Cedex, France
September 2011
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
1 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
1
2
3
Scalar conservation laws
Conclusion 1D Lagrangian hydrodynamics
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
1 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
Scalar conservation laws
Conclusion 1D Lagrangian hydrodynamics
extension of finite volumes method polynomial approximation of the solution in the cells high order scheme, high precision local variational formulation choice of the numerical fluxes (global L2 stability, entropic 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 September 2011
Franc¸ois Vilar
Cell-centered DG scheme
2 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
1
2
3
Scalar conservation laws
Conclusion 1D Lagrangian hydrodynamics
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
2 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
Conclusion
Scalar conservation laws
1D Lagrangian hydrodynamics
comparison between the second order and the third order scheme with limitation solution 2nd order 3rd 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
Figure: linear advection of a combination of smooth and discontinuous profiles September 2011
Franc¸ois Vilar
Cell-centered DG scheme
3 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
1D Lagrangian hydrodynamics
Burgers
advection : solid body rotation 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
0.1
0.2
0.3
0.4
0.5
0.6
Conclusion
Scalar conservation laws
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
numerical solutions using third order limited DG on a polygonal grid made of 2500 cells
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
4 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
Conclusion
Scalar conservation laws
1D Lagrangian hydrodynamics
rate of convergence with and without the slope limitation
linear advection
first order second order second order lim third order third order lim
L1 1.02 1.99 2.15 2.98 3.45
L2 1.02 1.98 2.15 2.98 3.22
Table: for the smooth solution u0 (x) = sin(2πx) sin(2πy ) on a [0, 1]2 Cartesian grid
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
5 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
1
2
3
Scalar conservation laws
Conclusion 1D Lagrangian hydrodynamics
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
5 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
Conclusion
Scalar conservation laws
1D Lagrangian hydrodynamics
influence of the limitation on the linearized Riemann invariants 1.1
0.3 solution 3rd order
solution 3rd order 3rd order limited
1 0.29 0.9 0.28
0.8
0.7
0.27
0.6 0.26 0.5 0.25
0.4
0.3 0.24 0.2 0.23 0.1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a) without limitation
0.8
0.9
1
0.65
0.7
0.75
0.8
0.85
0.9
(b) with and without limitation
Figure: third order DG for the Sod shock tube problem using 100 cells: density
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
6 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
Conclusion
Scalar conservation laws
1D Lagrangian hydrodynamics
3rd order DG scheme with limitation: density 1.8
5
solution 3rd order limited
solution 3rd order limited 4.5
1.7
4
1.6
3.5
1.5
3
1.4
2.5
1.3
2
1.2
1.5
1.1
1
1
0.9
0.5 -5
-4
-3
-2
-1
0
1
2
3
(a) Shu oscillating shock tube
4
5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(b) uniformly accelerated piston
P.-H. M AIRE, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes J. Comp. Phys., 2009 September 2011
Franc¸ois Vilar
Cell-centered DG scheme
7 / 30
Introduction
2D Lagrangian hydrodynamics
Discontinuous Galerkin (DG)
Conclusion
Scalar conservation laws
1D Lagrangian hydrodynamics
rate of convergence with and without the slope limitation
gas dynamics
first order second order second order lim third order third order lim
L1 0.80 2.25 2.04 3.39 2.75
L2 0.73 2.26 2.21 3.15 2.72
Table: for a smooth solution in the special case γ = 3
F. V ILAR , P.-H. M AIRE , 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
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
8 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
1
2
3
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
8 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
F. L. A DESSIO, D. E. C ARROLL , J. K. D UKOWICZ , N. L. J OHNSON , B. A. K ASHIWA , M. E. M ALTRUD, H. M. RUPPEL, Caveat: a computer code for fluid dynamics problems with large distortion and internal slip, Los Alamos National Laboratory, 1986 ´ R. L OUB E` RE, Une Methode Particulaire Lagrangienne de type ´ Galerkin Discontinu. Application a` la Mecanique des Fluides et l’Interaction Laser/Plasma, PhD thesis, Universite´ Bordeaux I, 2002 B. D ESPR E´ S , C. M AZERAN, Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems, Arch. Rational Mech. Anal., 2005 P.-H. M AIRE , R. A BGRALL , J. B REIL , J. OVADIA, A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comp, 2007 September 2011
Franc¸ois Vilar
Cell-centered DG scheme
9 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
1
2
3
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
9 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
gas dynamics system in Lagrangian formalism d 1 ( ) − ∇X (JF−1 U) = 0 dt ρ dU ρ0 + ∇X (JF−t P) = 0 dt dE ρ0 + ∇X (JF−1 PU) = 0 dt ρ0
(1a) (1b) (1c)
where X is the Lagrangian (initial) coordinate ∂x is called the deformation gradient tensor, where x is the F= ∂X Eulerian (actual) coordinate and J = det(F) using the trajectory equation
dF dx = U(x, t) =⇒ = ∇X U dt dt
Piola compatibility condition ∇X (JF−t ) = 0 September 2011
Franc¸ois Vilar
Cell-centered DG scheme
(2) (3) 10 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
1
2
3
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
10 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
being given a mapping x = Φ(X , t) F = ∇X Φ
(4)
developing Φ on the basis functions λp in the cell Ωc Φch (X , t) = Φh (X , t)|Ωc X = λp (X ) Φp (t) p
where the p points are some control points by setting Gc = (JF−t )c X ΦY (∂YX λp − ∂XY λp ) p ∇X Gc = =0 −ΦXp (∂YX λp − ∂XY λp ) p
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
11 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
using (4) and
d Φp = U p dt
=⇒
X d Fc = U p ⊗ ∇X λp dt p
(5)
in 2D, F −→ JF−t = G is a linear function JF−t N represents the geometric normal in the Eulerian frame thanks to Nanson formula JF−t NdS = GNdS = nds to ensure this quantity to be continuous, we discretize F by means of mapping defined on triangular cells Ti c with i = 1 . . . ntri, using finite elements polynomial basis
Ωc Tic
d dt F
= ∇X U, F approximation order has to be one 1 less than the one obtain with the DG scheme on , U and E ρ
using the fact
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
12 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
1
2
3
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
12 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
for the P1 representation, the chosen finite elements polynomial basis in a general triangle Tc write 1 λp (X ) = [X (Yp+ − Yp− ) − Y (Xp+ − Xp− ) + Xp+ Yp− − Xp− Yp+ ] (6) 2|Tc | we can access to ∇X λp needed in (5) 1 1 Yp+ − Yp− ∇X λp (X ) = Lpc N pc = 2|Tc | Xp− − Xp+ |Tc |
(7)
LpcN pc (Xp, Yp)
Lp− p N p− p + Lpp+ N pp+ 2 Lp+ p− N p+ p− =− 2
where Lpc N pc =
Tc (X, Y )
(Xp− , Yp− )
(Xp+ , Yp+ ) Lp + p − N p + p −
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
13 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
the equation (5) rewrites
d 1 Fc = dt |Tc |
X
U p ⊗ Lpc N pc
p∈P(Tc )
(8)
p+ T pp+
with this definition, GN continuity is well preserved at the interface betweens triangles
Tc p−
Gc Lpp+ N pp+
1 = |Tc | 1 = |Tc | =
September 2011
X
Lpt c Lpp+
pt ∈P(Tc )
X
(Lpp+ T pp+ Lpt c N pt c )
pt ∈P(Tc )
Franc¸ois Vilar
!
=
p
X NY − NY NX ) ΦYp (Npp + pt c pp+ pt c X NY − NY NX ) −ΦXp (Npp + pt c pp+ pt c
ΦYp+ − ΦYp ΦXp − ΦXp+
N pp+
yp+ − yp xp − xp+
ΦYp −ΦXp
!
= lpp+ npp+
Cell-centered DG scheme
(9) 14 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
1
2
3
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
14 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
Discontinuous Galerkin {σkc }k =0...K basis of Porder −1 (Ωc ) φch (X , t) =
K X
φck (t)σkc (X ) approximate of φ(X , t) on Ωc
k =0
Taylor basis, k1 + k2 = k σkc =
X − Xc k1 Y − Yc k2 X − Xc k1 Y − Yc k2 1 [( ) ( ) − h( ) ( ) i] k1 !k2 ! ∆Xc ∆Yc ∆Xc ∆Yc
for the second order scheme, K = 2 σ0c = 1, σ1c =
X − Xc c Y − Yc , σ2 = ∆Xc ∆Yc
min min where ∆Xc = Xmax −X and ∆Yc = Ymax −Y with Xmax , Ymax , 2 2 Xmin , Ymin the maximum and minimum coordinates in the cell Ωc September 2011
Franc¸ois Vilar
Cell-centered DG scheme
15 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
Density local variational formulation of (1a) on Ωc Z
ρ0
Ωc
Z K X d 1 d 1 ( )σq dΩ = ( )k ρ0 σq σk dΩ dt ρ dt ρ Ωc k =0 Z = σq ∇X (JF−1 U)dΩ Ωc Z Z = − U JF−t ∇X σq dΩ + Ωc
U σq JF−t NdL
∂Ωc
Gci = (JF−t )ci is constant on Ti c and ∇X σq over Ωc Z
ntri
ρ0
Ωc
X d 1 ( )σq dΩ = − Gci ∇X σq dt ρ
September 2011
i=1
Franc¸ois Vilar
Z
Z UdT +
Ti c
U σq GNdL ∂Ωc
Cell-centered DG scheme
16 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
Z ntri X d 1 c Gi ∇X σq UdT ρ ( )σq dΩ ' − Ti c Ωc dt ρ i=1 Z X + Up σq GNdL
Z
p
0
Ωpc p−
p+
Ωc
∂Ωc ∩∂Ωpc
p∈P(Ωc )
|
{z
q
}
q
lpc npc
finally, the equation on the density leads to Z
ntri
ρ0
Ωc
X d 1 ( )σq dΩ = − Gci ∇X σq dt ρ i=1
Z UdT + Ti
c
X
q q npc U p lpc
(10)
p∈P(Ωc )
0 n0 for the first order with lpc npc = lpc pc
d 1 mc ( )c = dt ρ September 2011
Franc¸ois Vilar
Z
ρ0
Ωc
X d 1 ( )dΩ = U p lpc npc dt ρ
(11)
p∈P(Ωc )
Cell-centered DG scheme
17 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
Velocity local variational formulation of (1b) on Ωc leads to Z
ntri
X U Gci ∇X σq ρ σq dΩ = dt Ωc 0d
i=1
where F qpc =
Z PdT − Ti
c
X
F qpc
(12)
p∈P(Ωc )
Z P σq GNdL ∂Ωc ∩∂Ωpc
for the first order with F pc = F 0pc d Uc mc = dt
September 2011
Franc¸ois Vilar
Z
ρ0
Ωc
X dU dΩ = − F pc dt
(13)
p∈P(Ωc )
Cell-centered DG scheme
18 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
Energy local variational formulation of (1c) on Ωc Z Z ntri X X Z dE Gci ∇X σq P UdT − σq dΩ = P U σq GNdL (14) ρ0 Ti c Ωc dt ∂Ωc ∩∂Ωpc i=1
p∈P(Ωc )
we make the following fundamental assumption
PU =PU
finally, the equation on the energy rewrites Z Z ntri X X dE ρ0 σq dΩ = Gci ∇X σq P UdT − U p F qpc c dt Ωc Ti i=1
(15)
p∈P(Ωc )
for the first order d Ec mc = dt September 2011
Franc¸ois Vilar
Z
ρ0
Ωc
X dE dΩ = − U p F pc dt
(16)
Cell-centered DG scheme
19 / 30
p∈P(Ωc )
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
Entropic analysis the use of variational formulations and Gibbs formula leads to Z Z 0 dS dΩ = [P U + P U − P U − P U] GNdL ρ T dt ∂Ωc Ωc X Z (17) = (P − P)(U − U) GNdL f ∈F (Ωc ) f
Z a sufficient condition to satisfy
ρ0 T
Ωc
dS dΩ ≥ 0 consists in setting dt
P(X f ) = Pc(X f ) − Zc (U(X f ) − U c (X f ))
GN kGNk
(18)
where X f is a point on the face f and Zc a positive constant with a physical dimension of a density times a velocity September 2011
Franc¸ois Vilar
Cell-centered DG scheme
20 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
using this expression to calculate F qpc leads to Z q P σq JF−t NdL F pc = ∂Ωc ∩∂Ωpc
Z
Z
GN σq GNdL Pc σq GNdL − Zc (U − U c ) kGNk ∂Ωc ∩∂Ωpc ∂Ωc ∩∂Ωpc Z ' Pc (p) σq GNdL =
∂Ωc ∩∂Ωpc
Z −
Zc (U p − U c (p)) ∂Ωc ∩∂Ωpc
GN σq GNdL kGNk
finally, F qpc writes q q F qpc = Pc (p) lpc npc − Mqpc (U p − U c (p))
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
(19)
21 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
Mqpc are defined as Mqpc = Zc p
Ω+ pc
=
Ω− pc
Z
∂Ωc ∩∂Ωpc q,+ + Zc (lpc npc ⊗
p−
p+
where
Ωc
q,± lpc
GN ⊗ GN σq dL kGNk q,− − − n+ pc + lpc npc ⊗ npc )
Z =
σq dL ∂Ωc ∩∂Ω± pc
+ + − − − M0pc = Mpc = Zc (lpc npc ⊗ n+ pc + lpc npc ⊗ npc ) is semi definite positive matrix with a physical dimension of a density times a velocity
toX be conservative in total energy over the whole domain, F pc = 0 and consequently c∈C(p)
(
X
Mpc ) U p =
c∈C(p) September 2011
X
[Pc (p) lpc npc + Mpc U c (p)]
(20)
c∈C(p) Franc¸ois Vilar
Cell-centered DG scheme
22 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
1.1 exact solution 1st order 2nd order
1
0.7
1
0.9 0.6
0.9
0.8 0.8
0.5 0.7
0.7
0.4
0.6 0.5
0.3
0.2
0.6 0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
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
Sod shock tube problem on a polar grid made of 500 cells: density map with limitation
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
23 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
3 exact solution 2nd order
2.4 2
2.2
2.5
2 2
1.5
1.8 1.6
1
1.5
1.4 1.2
0.5
1
1 0.5
0.8 0.6
0 0
0.5
1
1.5
2
2.5
3
0 0
0.5
1
1.5
2
2.5
3
3.5
expansion wave into vacuum problem on a polar grid made of 250 cells: internal energy map with limitation
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
24 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
1st order
2nd order limited 16
16 0.5
0.5
14
14 12 0.4
0.4 12 10 10
0.3
0.3 8
8 0.2
0.2 6 6
0.1
4
4
0.1
2
2 0
0 0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
Noh problem on a Cartesian grid made of 2500 cells: density map
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
25 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
5.5
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
Sedov problem on a Cartesian grid made of 900 cells and a polygonal one made of 775 cells: density map with limitation
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
26 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
initial grid
actual grid
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Gresho problem on a polar grid made of 720 cells: pressure map with limitation September 2011
Franc¸ois Vilar
Cell-centered DG scheme
27 / 30
1
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
discontinuous Galerkin
discontinuous Galerkin limited
1
1.4
0.9
1.3
0.8
1 1.3
0.9
0.8
1.2
1.2 0.7
0.7
1.1
1.1 0.6
0.6 1
1 0.5
0.5 0.9
0.9 0.4
0.4 0.8
0.3
0.8 0.3
0.2
0.7
0.2
0.7
0.1
0.6
0.1
0.6
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Taylor-Green vortex problem on a cartesian grid made of 400 cells: pressure map without limitation at t=0.75s
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
28 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme
h 1 20 1 40 1 80
without limitation qLh2 qLh∞ 1.74 1.35 1.85 1.85 1.42 2.34
with limitation qLh2 qLh∞ 2.05 1.54 2.11 1.81 1.58 1.54
Table: rate of convergence computed for second order DG scheme
h 1 20 1 40 1 80 1 160
Green Muscl ELh2 ELh∞ 1.854E-2 6.596E-2 6.500E-3 2.452E-2 1.817E-3 9.122E-3 4.944E-4 2.555E-3
Discontinuous Galerkin ELh2 ELh∞ 1.120E-2 3.678E-2 3.356E-3 1.446E-2 9.314E-4 4.019E-3 3.471E-4 7.959E-4
Table: numerical errors computed at t=0.6s on the pressure
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
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Introduction
1
2
3
2D Lagrangian hydrodynamics
Conclusion
Introduction Discontinuous Galerkin (DG) Scalar conservation laws 1D Lagrangian hydrodynamics 2D Lagrangian hydrodynamics References System and equations Geometric consideration 2nd order Deformation tensor 2nd order DG scheme Conclusion
September 2011
Franc¸ois Vilar
Cell-centered DG scheme
29 / 30
Introduction
2D Lagrangian hydrodynamics
Conclusion
Conclusions DG schemes up to 3rd order linear and non-linear scalar conservation laws in 1D and 2D on general unstructured grids 1D gas dynamics system in Lagrangian formalism
DG scheme up to 2nd order for the 2D gas dynamics system in Lagrangian formalism with particular geometric consideration numerical flux studies Riemann invariants limitation
Prospects 3rd order DG scheme for the 2D gas dynamics system in Lagrangian formalism validation extension to ALE September 2011
Franc¸ois Vilar
Cell-centered DG scheme
30 / 30
