A DG method for Lagrangian hydrodynamics

1.2 Numerical flux and L. 2 stability goal: access to the L. 2 norm of our solution and insure stability. Mono-dimensional problems: • f is integrable and its ...
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A discontinuous Galerkin method for Lagrangian hydrodynamics F. Vilar, P.-H. Maire and R. Abgrall CELIA, Universit´e Bordeaux I, CNRS, CEA 1 1.1

2

Discontinuous Galerkin (DG) introduction with scalar conservation laws 2.1

Discretization ∂u ∂f (u) + =0 ∂t ∂x

•{ej }j=1..K a basis of our approximation space PK (Ci) and uih =

K X

∂tuil

l=0

Z

Ci

xi+ 1 i i i (el , ek )dx + [f (u)ek ]xi− 21 −

Z

dt du 0 ρ dt dE 0 ρ dt

2

Ci

2.2 f (uih)∂xeik dx = 0

R

h ix 1 d i+ 2 i i i − D iF i = 0 M U + f (u)(x)B (x) dt xi− 1 2   1 x − xi k x − xi k i We could use local Taylor basis {ej }j=1..K where ek = ( ) −( ) , xi is the centroid of the cell Ci. k! ∆xi ∆xi

Acoustic

• linearization by small perturbations of the gas dynamics system, around a steady flow ⇒ (p, u) system 0.6

1.1 solution 1st order 2nd order 3rd order

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solution p and u limitation Riemann invariants limitation

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Influence of orders 2.2.2

goal: access to the L2 norm of our solution and insure stability

• f is integrable and its derivative smooth enough as F (u) =

Z u

f (s)ds

2

Shallow water

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3) Ci+ 1 =

a2 ∆t

2

2 ∆xi

: Lax-Wendroff

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3 solution 3rd order

solution 3rd order

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1

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3rd order DG for a oscillating tube problem

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3rd order DG for a uniformly 3rd order DG for a shock 3rd order DG for a shock Sod accelerated piston problem Sod tube problem: density tube problem: internal energy

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We see that the oscillations are quite strong at the shock front, without any limitation. So, to keep our solution monotone, as we did for the shallow water equations, we linearize the system on each cells and obtain linear quantities on which we can perform our limitation. The problem is how can we limit our last unknown, E, the total energy. We have tested different ways but at the end, some little oscillations still remain.

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Numerical results

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L2

L∞

1st order

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0.94

2nd order

2.05

2.05

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2nd order lim

2.37

2.05

1.61

3rd order

3.00

3.00

2.89

3rd order lim

3.32

3.10

2.59

1st order

0.86

0.68

0.23

2nd order

2.00

1.99

1.91

2nd order lim

2.12

1.99

1.57

3rd order

2.88

2.91

2.65

3rd order lim

2.87

2.89

2.62

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Linear advection 0.6

3

1st order 2nd order 3rd order

1

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Burgers

0

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Influence of the orders on a linear advection problem

• Different physical problems, linears and nonlinears, were studied in a Lagrangian formalism and explicit formulas for the flux, to have L2 or entropy stability, have been shown

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• Difficulties residing in limiting nonlinear systems have been noticed • Multidimensional studies will be pursued for the Lagrangian hydrodynamics problem presented before

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Influence of the orders on a Burgers problem

Numerical order of our methods in 1D

Conclusions and perspectives

• Our DG methods have been validated and so, order influence on the accuracy was observed

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• Lagrangian schemes, using the initial mesh, will be studied, in order to avoid the cells deformation problem due to high orders

References 1

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Linear advection problem Burgers problem with a 3rd Buckley problem with a 3rd with a 3rd order DG order DG method on order DG method on method on polygonal cells polygonal cells polygonal cells

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−2

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2 1 −1.5

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KPP problem with a 3rd order DG method on polygonal cells

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solution 3rd order 3rd order limited

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L1

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Influence of the orders with limitation solution 1st order 2nd order 3rd order

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3rd order DG for a dam break problem: velocity

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encountered, we stop the limitation.

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To enforce monotonicity, we perform a vertex based slope limitation [2]. i 2nd order, we have uih = ui0 + αl ui1 x−x ∆x with αl ∈ [0, 1], the correction factor

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solution2 3rd order 1.7

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(1) (1) i x−xi (2) (2) ui1 ∂ i i i 3rd order, we set ui = u0 + αl u1 ∆x and ui = ∂x uh = ∆x + αl ui2 x−x . 2 ∆xi i i (1) In order to avoid the loss of accuracy at smooth extrema, we set αl = (1) (2) max(αl , αl ). For high order, we calculate a nondecreasing sequence of cor(q) (q) (p) rection factors αl = max(αl ), q ≥ p, that means, as soon as αl = 1 is 0.2

1.2

Gas dynamics

i

0

1

solution 3rd order

Z ud −−→ f (u)du − (ud − ug )Mfe − n→ fe

Limitation

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• we linearize, on each cells, the equations and thus, we obtain linear approximation of the Riemann invariants on each cells. Then, the limitation is easy.

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• these differentials being quite simple, we were able to integrate and differentiate them and so, perform our limitation on the Riemann invariants.

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−→ −→ 1 −→ −→ 1 −→ −→ Afe ⊗ Afe ) • Examples for linear advection: Mfe = |Afe .nfe |I upwind scheme or Mfe = |Afe .nfe |( −→ 2 2 ||Afe ||2

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As before, if we apply our limitation on the intrinsic system variables, some oscillations remain. The problem is this equations system is nonlinear, and so we have only informations on the differential of the Riemann invariants. In order to access to this quantities, we’ve tested two different options :

solution h and u limitation Riemann invariants limitation

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∆xi 2) Ci+ 1 = : Lax-Friedrichs, 2 2∆t

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3rd order DG for a rarefaction 3rd order DG for a double 3rd order DG for a dam wave into vacuum problem rarefaction waves problem break problem: water height

1 f (u) = ud − ug ug

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• same procedure and we find a similar expression for the numerical flux, on the face fe, with Mfe a positive definite matrix:

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• For linear advection, f (u)(xi+ 1 ) = a2 (ug + ud) − Ci+ 1 (ud − ug ):

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solution 3r order 0.9

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fe

solution 3rd order

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2

1

solution 3rd order

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⇒ we find a sufficient condition on f (u) with Ci+ 1 ≥ 0, to have Ri ≥ 0: 2 Z ud 1 f (u)(xi+ 1 ) = f (u)du − Ci+ 1 (ud − ug ) 2 2 ud − ug ug

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• permutation of the sum from cells to nodes and ug = uh(x− 1 ), ud = uh(x+ 1 ) i+ 2 i+ 2

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xi+ 1 xi+ 1 • sum on all cells with Ri = [f (u)uh]xi− 21 − [F (uh)]xi− 21

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1

xi+ 1 xi+ 1 u2h d 2 dx + [f (u)uh]xi− 1 − [F (uh)]xi− 21 = 0 dt Ci 2 2 2

solution 2nd order 3rd order

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solution 3rd order

Z

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Influence of limitations

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Multi-dimensional problems:

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• small water height, an incompressible fluid, a sliding condition at the bottom and average of the equations on the water height ⇒ (h, u) system where h is the water height

Mono-dimensional problems:

|a| 1) Ci+ 1 = : upwind, 2 2

We notice that if we just perform the limitation on the system unknowns, some oscillations remain. But, by diagonalizing the system, we get around this constraint. For the acoustic system, it is quite simple because these invariants can be found explicitly (due to the linear property of this system) but for the other cases, it isn’t so obvious.

1

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Numerical flux and L2 stability

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with ρ the density of the fluid, ρ0 its initial density, u its velocity and E its total energy. For a thermodynamic closure of this system, we introduce 2 u an equation of state p = p(ρ, ε) with ε = E − 2 . We may, for example, use the ideal gas law : p = ρ(γ − 1)ε.

∂X ∂p + = 0 ∂X ∂pu + = 0 ∂X

Numerical results

2.2.1

i = i , ei)dx, D i = i , ei)dx, B i(x) = (ei (x), ..., ei (x), ..., ei (x))T , F i = (f i, ..., f i , ..., f i )T , •Mkl (e (∂xe 0 0 Ci k l Ci K K kl k l l l U i = (ui0, ..., uil , ..., uiK )T our unknown vector

R

∂u d(1/ρ) 0 − = 0 ρ

uij (t)eij (x) our approximate solution on Ci

j=0

K X

Gas dynamics in Lagrangian formalism

goal: approximate our solution by polynomials on each cells without imposing continuity between them

u(x, 0) = u0(x)

Lagrangian hydrodynamics

[1] B. Cockburn, Discontinuous Galerkin methods School of Mathematics, University of Minesota, 2003. [2] D. Kuzmin A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods J. of Comp. Appl. Math. doi:10.1016/j.cam.2009.05.028. [3] B. Cockburn, S.-Y. Lin and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II : one-dimentional J. of Comp. Phys. 84:90-113, 1989. [4] B. Popov, Entropy viscosity MULTIMAT 2009, September 21-25 2009, Pavia, Italy [5] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II : general framework. Math. Comput. 52:411-435, 1989. [6] M. Yang and Z.J. Yang A parameter-free generalized moment limiter for high-order methods on unstructured grids Adv. Appl. Math. Mech. 1(4):451-480,2009.