Contribution à l approximation numérique des systèmes hyperboliques

(1)

Contribution à l’approximation numérique des systèmes hyperboliques

Vivien Desveaux

sous la direction de Christophe Berthon et Yves Coudière

Laboratoire de mathématiques Jean Leray - Université de Nantes

26 novembre 2013

(2)

1 Second-order schemes based on dual mesh gradient reconstruction

2 A high-order entropy preserving scheme witha posteriori limitation

3 Well-balanced schemes for systems with nonlinear source terms

(3)

1 Second-order schemes based on dual mesh gradient reconstruction MUSCL scheme and CFL condition

The DMGR scheme Numerical results

2 A high-order entropy preserving scheme witha posteriori limitation Motivations

From one to all discrete entropy inequalities The e-MOOD scheme for the Euler equations

3 Well-balanced schemes for systems with nonlinear source terms The Ripa model

Relaxation models The relaxation scheme Numerical results

(4)

Dual Mesh Gradient Reconstruction schemes

Introduction

Hyperbolic system of conservation laws in 2D

∂_tw+∂_xf(w) +∂_yg(w) = 0 w:R⁺×R²→Ω⊂R^d: unknown state vector f,g: Ω→R^d: flux functions

Ω convex set of physical states Objective: derive a numerical scheme

◮ second order accurate

◮ Ω–preserving

◮ works on unstructured meshes

◮ with an optimized CFL condition

(5)

Mesh notations

Kj3

Kj4

K_j5

Kj1

K_j2

Ki νij1

ν_ij2

νij³

νij⁴ νij5

eij1

Geometry of the cell Ki

polygonal cells Ki (perimeter Pi, area|K_i|)

γ(i): index set of the cells neighbouringK_i

e_ij: common edge between K_i and K_j (length|eij|)

ν_ij: unit outward normal to e_ij

(6)

Dual Mesh Gradient Reconstruction schemes MUSCL scheme and CFL condition

First-order scheme

w_iⁿ⁺¹ =w_iⁿ− ∆t

|Ki| X

j∈γ(i)

|e_ij|ϕw_iⁿ,w_jⁿ, ν_ij

2D numerical flux: ϕGodunov-type in each direction ν [Harten, Lax & van Leer ’83]:

ϕ(wL,wR, ν) =h_ν(wL) + δ

2∆twL− 1

∆t Z ₀

−^δ

2

we_ν x

∆t,wL,wR

dx

with respect to the CFL condition ∆t

δ maxλ^±(wL,wR, ν)≤ 1 2

◮ hν(w) =νxf(w) +νyg(w): flux in theν–direction, withν= (νx, νy)^T

◮ weν approximate Riemann solvervalued in Ω Consistency: ϕ(w,w, ν) =h_ν(w)

Conservation: ϕ(wL,wR, ν) =−ϕ(wR,wL,−ν)

(7)

Theorem (Robustness of the first-order scheme) Assume the following CFL condition is satisfied:

∆t P_i

|Ki| max

j∈γ(i)

λ^±(w_iⁿ,wⁿ_j, ν_ij)≤1, ∀i∈Z. Then the first-order scheme preserves Ω:

∀i ∈Z, w_iⁿ ∈Ω ⇒ ∀i ∈Z, w_iⁿ⁺¹ ∈Ω.

Remark: in [Perthame & Shu ’96], they obtain the CFL restriction

∆t P_i

|Ki| max

j∈γ(i)

λ^±(w_iⁿ,wⁿ_jν_ij)≤ 1

2, ∀i∈Z.

However this can easily be improved in the Godunov-type framework.

(8)

MUSCL scheme

([van Leer ’79], [Perthame & Shu ’96]...)

First-order scheme on the cell K_i w_iⁿ⁺¹ =w_iⁿ− ∆t

|Ki| X

j∈γ(i)

|e_ij|ϕw_iⁿ,w_jⁿ, ν_ij

Second-order MUSCL scheme on the cellK_i w_iⁿ⁺¹=wⁿ_i − ∆t

|Ki| X

j∈γ(i)

|eij|ϕ(wij,wji, νij)

wij and wji are second-order approxima- tions at the interface between Ki andKj

b

b b

wⁿ_i

wⁿ_j w_ij

w_ji Ki

Kj

eij

→ How to compute wij ?

(9)

Subcells decomposition

K_j3

K_j4

K_j₅ K_j₁ K_j₂

T_ij3

T_ij₄ T_ij₅ T_ij₁ T_ij2

e_jⁱ

1j5

ν_jⁱ₁_j₅

bG_i

Subcells decomposition of the cell Ki

Tij: triangle formed by the mass center G_i and the edgee_ij (perimeter Pij, area|Tij|) γ(i,j): index set of the two subcells neighbouringTij inKi

e_jkⁱ : common edge between T_ij and Tik (length|e_jkⁱ |)

ν_jkⁱ : unit outward normal toe_jkⁱ

(10)

Theorem (Robustness of the MUSCL scheme) Assume the following hypotheses:

(i) The reconstruction preserves Ω:

∀i∈Z, wⁿ_i ∈Ω ⇒ ∀i ∈Z, ∀j∈γ(i), w_ij ∈Ω.

(ii) The reconstruction satisfies the conservation property X

j∈γ(i)

|Tij|

|K_i|wij =w_iⁿ.

(iii) The following CFL condition is satisfied for all i ∈Z:

∆t max

j∈γ(i)

Pij

|Tij| max

k∈γ(i,j)

λ^±(wij,wji, νij), λ^±(wij,wik, ν_jkⁱ )≤1 Then the MUSCL scheme preserves Ω:

∀i ∈Z, w_iⁿ ∈Ω ⇒ ∀i ∈Z, w_iⁿ⁺¹ ∈Ω.

(11)

The DMGR scheme

b b b b

b

b b b b b

b

b b b b

b

b b

bb

× ×

×

× ×

×

× ×

×

b Vertex of the primal mesh (unknown state)

b Center of a primal cell

= Vertex of the dual mesh (known state)

× Center of a dual cell (known state)

Figure: Primal mesh (blue) and dual mesh (red)

1 We write a MUSCL scheme on both primal and dual meshes

⇒ At timetⁿ, we know a state at the center of each primal and dual cell

(12)

Dual Mesh Gradient Reconstruction schemes The DMGR scheme

The DMGR scheme

b b b b

b

b b b b b

b

b b b b

b

b b

bb

× ×

×

× ×

×

× ×

×

Figure: Primal mesh (blue) and dual mesh (red) Assume we have a reconstruction procedure on a generic cell K:

(state at the center

states at the vertices 7→ linear reconstruction w_e This procedure will be detailed after.

(13)

The DMGR scheme

b b b b

b

b b b b b

b

b b b b

b

b b

bb

× ×

×

× ×

×

× ×

×

Figure: Primal mesh (blue) and dual mesh (red)

2 We can apply the reconstruction procedure on the dual cells

⇒ We get a linear functionw_e_i^d on each dual cell

3 To get the state at the primal vertexS_i^p, we takew_e_i^d(S_i^p)

⇒ We can apply the reconstruction procedure on the primal cells to get a linear functionwe_i^p

(14)

Reconstruction procedure

T7/2

T9/2 T1/2

T3/2

T5/2

b b b

b

b b b

b

b b

S1

S2

S3

S4

S5

G

Q3/2

Q5/2

Q7/2

Q9/2 Q1/2

Geometry of the cell K

b b b

b

b b b

b

b b

w1

w2

w3

w4

w5

w0

b w3/2 b

w5/2

b w7/2

b

w9/2 wb1/2

Known statesand reconstructed states

The states wb_j−1/2 have to satisfy: w_b_j−1/2 ∈Ω X

j

|T_j−_1/2|

|K| w^b_j−1/2 =w₀

If we take w_b_j−_1/2 =w(Q_e _j−_1/2) with w_e a linear function onK, we have X

j

|T_j−_1/2|

|K| w^b_j−1/2 =w₀ ⇔ w(G) =e w₀

(15)

Reconstruction procedure

T7/2

T9/2 T1/2

T3/2

T5/2

b b b

b

b b b

b

b b

S1

S2

S3

S4

S5

G

Q3/2

Q5/2

Q7/2

Q9/2 Q1/2

b b b

b

b b b

b

b b

w1

w2

w3

w4

w5

w0

b w3/2 b

w5/2

b w7/2

b

w9/2 wb1/2

1 Gradient reconstruction

We define a continuous functionw:K →R^d piecewise linear on each triangle T_j−1/2 and such that w(S_j) =w_j and w(G) =w₀.

(16)

Reconstruction procedure

T7/2

T9/2 T1/2

T3/2

T5/2

b b b

b

b b b

b

b b

S1

S2

S3

S4

S5

G

Q3/2

Q5/2

Q7/2

Q9/2 Q1/2

b b b

b

b b b

b

b b

w1

w2

w3

w4

w5

w0

b w3/2 b

w5/2

b w7/2

b

w9/2 wb1/2

2 Projection

For a slopeα∈M_d,2(R), we definew_e_α(X) =w₀+α·(X −G), the linear function whose gradient isα.

Letµ be the slope resulting from theL²–projection ofw:

Z

K

kw(X)−weµ(X)k²dX = min

α∈Md,2(R)

Z

K

kw(X)−weα(X)k²dX.

(17)

Reconstruction procedure

T7/2

T9/2 T1/2

T3/2

T5/2

b b b

b

b b b

b

b b

S1

S2

S3

S4

S5

G

Q3/2

Q5/2

Q7/2

Q9/2 Q1/2

b b b

b

b b b

b

b b

w1

w2

w3

w4

w5

w0

b w3/2 b

w5/2

b w7/2

b

w9/2 wb1/2

3 Limitation of the slopeµ

We define the optimal slope limiter by:

α_j−_1/2 = supⁿθ∈[0,1],w_e_sµ(Q_j−_1/2)∈Ω,∀s∈[0, θ]^o, β = min

j α_j−_1/2−ǫ,

whereǫ >0 is a small paramater s.t. we_βµ(Q_j−1/2)∈Ω, ∀j.

(18)

Reconstruction procedure

T7/2

T9/2 T1/2

T3/2

T5/2

b b b

b

b b b

b

b b

S1

S2

S3

S4

S5

G

Q3/2

Q5/2

Q7/2

Q9/2 Q1/2

b b b

b

b b b

b

b b

w1

w2

w3

w4

w5

w0

b w3/2 b

w5/2

b w7/2

b

w9/2 wb1/2

4 Finally, the reconstructed states are given by wb_j−1/2 =w_e_βµ(Q_j−1/2).

Limitation procedure ⇒ w_b_j−1/2 ∈Ω

w(G) =e w₀ ⇒ ^X

j

|T_j−1/2|

|K| w^b_j−1/2=w₀

⇒ The DMGR scheme preserves Ω

(19)

Numerical results: 2D Euler equations

∂t





 ρ ρu ρv E





+∂x





 ρu ρu²+p

ρuv u(E+p)





+∂y





 ρv ρuv ρv²+p v(E +p)





= 0

ρ: density (u,v): velocity E: total energy

p: pressure given by the ideal gas law

p= (γ−1)

E−ρ 2

u²+v², withγ ∈(1,3]

Set of physical states Ω =

(ρ, ρu, ρv,E)∈R⁴; ρ >0, E−ρ 2

u²+v²>0

(20)

Dual Mesh Gradient Reconstruction schemes Numerical results

2D Riemann problems

Figure: Four shocks 2D Riemann problem on a Cartesian mesh with 1.5×10⁶DOF

Figure: Four contact discontinuities 2D Riemann problem on a Cartesian mesh with 1.5×10⁶ DOF

(21)

Double Mach reflection on a ramp

Figure: Double Mach reflection on a ramp on an unstructured mesh with 3×10⁶ DOF

(22)

Dual Mesh Gradient Reconstruction schemes Numerical results

Mach 3 wind tunnel with a step

Figure: Mach 3 tunnel with a step on an unstructured mesh with 1.5×10⁶ DOF

(23)

(24)

A high-order entropy preserving scheme

Introduction

Hyperbolic system of conservation laws in 1D

∂_tw+∂_xf(w) = 0 w:R⁺×R→Ω⊂R^d: unknown state vector f : Ω→R^d: flux function

Ω convex set of physical states Entropy inequalities:

∂tη(w) +∂xG(w)≤0, wherew 7→η(w) is convex and∇_wf∇_wη =∇_wG Objectives:

◮ Study the entropy stability of high-order schemes

◮ Derive a high-order numerical scheme for the Euler equations which is entropy preserving in the sense of Lax-Wendroff

(25)

Scheme notations

Space discretization: cellsKi = [x_i−1/2,x_i+1/2] with constant size

∆x=x_i+1/2−x_i−1/2

w_iⁿ: approximate solution at time tⁿ on the cell K_i Update at time tⁿ⁺¹=tⁿ + ∆t given by

∆x

F_i+1/2ⁿ −F_iⁿ−1/2

whereF_i+1/2ⁿ =F w_iⁿ−s+1,· · · ,w_i+sⁿ andF is a consistant numerical flux (F(w,· · · ,w) =f(w))

We introduce the piecewise constant function

w^∆(x,t) =w_iⁿ, for (x,t)∈Ki×[tⁿ,tⁿ⁺¹)

The sequence (∆x,∆t) is devoted to converge to (0,0), the ratio

∆t

∆x being kept constant.

(26)

A high-order entropy preserving scheme Motivations

Lax-Wendroff Theorem

Theorem

(i) Assume the following hypotheses:

There exists a compact K ⊂Ωsuch that w^∆∈K ; w^∆ converges in L_loc¹ (R×R⁺; Ω) to a function w.

Then w is a weak solution.

(ii) Assume the additional hypothesis:

For all entropy pair(η,G), there exists an entropy numerical flux G, consistant with G (G(w,· · ·,w) =G(w)), such that we have the discrete entropy inequality

η(wⁿ⁺¹_i )−η(wⁿ_i)

∆t +G_i+1/2ⁿ −G_i−1/2ⁿ

∆x ≤0,

with G_i+1/2ⁿ =G w_i−s+1ⁿ ,· · ·,w_i+sⁿ . Then w is an entropic solution.

(27)

Example: the MUSCL scheme

Limited slope µⁿ_i =L w_iⁿ−w_iⁿ−1,w_i+1ⁿ −w_iⁿ, with L a limiter The MUSCL flux is defined by

F_i+1/2ⁿ =F w_iⁿ+µⁿ_i/2,w_i+1ⁿ −µⁿ_i+1/2

The known discrete entropy inequalities satisfied by the MUSCL scheme all write

η(w_iⁿ⁺¹)−η(w_iⁿ)

∆t +G_i+1/2ⁿ −G_iⁿ₋_1/2,

∆x ≤ P_iⁿ −η(w_iⁿ)

∆t whereP_iⁿ =P(w_iⁿ, µⁿ_i,∆x, η).

Examples of operator P:

P₁(w, µ,∆x, η) = 1

∆x Z _∆x/2

−∆x/2

η

w+ x

∆xµ

dx [Bouchut et al. ’96]

P₂(w, µ,∆x, η) = η(w−µ/2) +η(w+µ/2)

2 [Berthon ’05]

(28)

Convergence study

The discrete entropy inequality η(w_iⁿ⁺¹)−η(w_iⁿ)

∆t +G_i+1/2ⁿ −G_iⁿ₋_1/2,

∆x ≤ P_iⁿ −η(w_iⁿ)

∆t converges weakly to

∂tη(w) +∂xG(w)≤δ, whereδ is a positive measure.

Conjecture (Hou-LeFloch ’94)

δ= 0 in the areas where w is smooth δ >0 on the curves of discontinuity ofw

(29)

Numerical study: test cases (Euler equations)

Entropy error (total mass of the right-hand side):

I^∆= ∆x^X

i,n

(P_iⁿ−η(w_iⁿ))

1–rarefaction

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

I^{∆ ?}−→0

Double shock

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 1

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

I^{∆ ?}−→c>0

(30)

Numerical results obtained with a second-order time scheme

1-rarefaction

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

∆ x

I∆

Superbee MC van Leer van Albada 1 minmod

Double shock

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

0 2 4 6 8 10 12 14

∆ x

I∆

Superbee MC van Leer van Albada 1 minmod

(31)

Conclusion of motivations

Numerical results confirm the Hou-le Floch conjecture: when the scheme converges, the measure δ seems to be concentrated on the curves of discontinuity of w.

This does not imply that the limit is not entropic, but the usual discrete entropy inequalities are not relevant to apply the Lax-Wendroff theorem.

We have to enforce the stronger discrete entropy inequalities η(w_iⁿ⁺¹)−η(w_iⁿ)

∆t +G_i+1/2ⁿ −G_iⁿ₋_1/2

∆x ≤0.

We suggest to extend thea posteriorimethods (MOOD) introduced in [Clain, Diot & Loubère ’11].

(32)

A high-order entropy preserving scheme From one to all discrete entropy inequalities

The family of entropies for the Euler equations

Lemma

The entropy pairs (η,G) of the Euler system write η=ρψ(r), G=ρψ(r)u,

where r =−^p^1/γ_ρ and ψ is a smooth increasing convex function.

We consider the scheme

∆x

F_i+1/2−F_i−1/2

,

where w_iⁿ = (ρⁿ_i, ρⁿ_iu_iⁿ,E_iⁿ)^T and F_i+1/2 = (F_i+1/2^ρ ,F_i+1/2^ρu ,F_i+1/2^E )^T.

(33)

We introducer_i+1/2ⁿ =

( r_i+1ⁿ ifF_i+1/2^ρ <0 r_iⁿ ifF_i+1/2^ρ >0 . Theorem

Assume the scheme preserves Ω. Assume the scheme satisfies the specific discrete entropy inequality

ρⁿ⁺¹_i r_iⁿ⁺¹≤ρⁿ_ir_iⁿ − ∆t

∆x

F_i+1/2^ρ r_i+1/2ⁿ −F_i^ρ₋_1/2r_iⁿ₋_1/2.

Assume the additional CFL like condition

∆t

∆x

max0,F_i+1/2^ρ −min0,F_i^ρ₋_1/2≤ρⁿ_i.

Then the scheme is entropy preserving: for all smooth increasing convex function ψ, we have

ρⁿ⁺¹_i ψ(r_iⁿ⁺¹)≤ρⁿ_iψ(r_iⁿ)− ∆t

∆x

F_i+1/2^ρ ψ(r_i+1/2ⁿ )−F_i^ρ₋_1/2ψ(r_iⁿ−1/2).

(34)

A high-order entropy preserving scheme The e-MOOD scheme for the Euler equations

First-order scheme

We consider a first-order scheme w_iⁿ⁺¹ =w_iⁿ− ∆t

∆x F w_iⁿ,wⁿ_i+1−F w_iⁿ−1,w_iⁿ. For a time step restricted according to the CFL condition

∆t

∆xmax

i∈Z

λ^± w_iⁿ,w_i+1ⁿ ≤ 1 2, the first-order scheme is assumed to satisfy:

Robustness: ∀i ∈Z, w_iⁿ ∈Ω ⇒ ∀i ∈Z, w_iⁿ⁺¹ ∈Ω Stability:

ρⁿ⁺¹_i r_iⁿ⁺¹≤ρⁿ_ir_iⁿ− ∆t

∆x

F^ρ w_iⁿ,w_i+1ⁿ r_i+1/2ⁿ

−F^ρ w_iⁿ−1,w_iⁿr_iⁿ−1/2

.

Example: the HLLC/Suliciu relaxation scheme

(35)

Reconstruction procedure

We consider high-order reconstructed states w_i^n,^± on the cell K_i at the interfaces x_i±1/2.

These reconstructed states can be obtained by any reconstruction procedure (MUSCL, ENO/WENO, PPM, DMGR...).

Assumptions:

◮ The reconstruction is Ω-preserving: w_i^n,±∈Ω;

◮ The reconstruction is conservative:

w_iⁿ =1

2 w_i^n,−+w_i^n,+

.

(36)

The e-MOOD algorithm

1 Reconstruction step: For alli ∈Z, we evaluate high-order reconstructed statesw_i^n,^± located at the interfaces x_i±1/2.

2 Evolution step: The solution is evolved as follows:

w_i^n+1,⋆=w_iⁿ − ∆t

∆x

Fw^n,+_i ,w^n,_i+1⁻−Fw^n,+_i₋₁,w_i^n,⁻.

3 A posteriori limitation step: We have the following alternative:

◮ if for alli∈Z, we have

ρ^n+1,⋆r_i^n+1,⋆≤ρⁿ_ir(w_iⁿ)− ∆t

∆x

F^ρ w_i^n,+,w_i+1^n,⁻ r_i+1/2ⁿ

−F^ρ w_i^n,+−1,w_i^n,⁻ r_i−1/2ⁿ

, (1) then the solution is valid and the updated solution at timetⁿ+ ∆t is defined byw_iⁿ⁺¹=w_i^n+1,⋆;

◮ otherwise, for alli∈Zsuch that (1) is not satisfied, we set wi^n,^±=wiⁿ and we go back to step 2.

(37)

Theorem

Assume the time step ∆t is chosen in order to satisfy the two following CFL like conditions:

∆t

∆xmax

i∈Z

λ^±w_i^n,+,w_i+1^n,⁻,λ^±w_i^n,⁻,w_i^n,+≤ 1 4,

∆t

∆x

max0,F_i+1/2^ρ −min0,F_i−^ρ _1/2≤ρⁿ_i.

Then the updated states w_iⁿ⁺¹, given by the e-MOOD scheme, belong to Ω. Moreover, for all smooth increasing convex function ψ, the e-MOOD scheme satisfies

1

∆t

ρⁿ⁺¹_i ψ(r_iⁿ⁺¹)−ρⁿ_iψ(r_iⁿ)+ 1

∆x

F^ρw_i^n,+,w_i+1^n,⁻ψ(r_i+1/2ⁿ )

−F^ρw_i^n,+₋₁,w^n,_i ⁻ψ(r_iⁿ₋_1/2)≤0.

The e-MOOD scheme is thus entropy preserving.

(38)

Numerical results obtained with a second-order time scheme

L¹ error: _X

i

ρ^N_i −ρex(xi,T) 1-rarefaction

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

∆ x

Erreur L1

Superbee minmod e−MOOD Superbee e−MOOD minmod

Double shock

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

∆ x

Erreur L1

Superbee minmod e−MOOD Superbee e−MOOD minmod

(39)

(40)

Well-balanced relaxation schemes The Ripa model

The Ripa model









∂th+∂xhu= 0

∂thu+∂x hu²+gh²θ/2=−ghθ∂xZ

∂_thθ+∂_xhθu= 0 h: water height

u: velocity θ: temperature g: gravity constant

Z(x): smooth topography function

(41)

The Ripa model









∂th+∂xhu= 0

∂_thθ+∂_xhθu= 0 We define

◮ the vector of conservative variablesw= (h,hu,hθ)^T,

◮ the flux functionf(w) = (hu,hu²+gh²θ/2,hθu)^T,

◮ the source term s(w) = (0,−ghθ,0)^T, to rewrite the system into the compact form

∂tw+∂xf(w) =s(w)∂xZ. The set of physical admissible states is

Ω =ⁿw∈R³, h >0, θ >0^o.

(42)

Well-balanced relaxation schemes The Ripa model

The Ripa model









∂th+∂xhu= 0

∂_thθ+∂_xhθu= 0

Steady states

The steady states at rest are governed by the ODE (u ≡0,

∂x(h²θ/2) =−hθ∂xZ.

We cannot obtain an explicit expression of all the steady states.

Lake at rest type solutions







 u = 0, θ= cst, h+Z = cst,







 u = 0, z = cst, h²θ= cst,







 u = 0, h = cst,

z+^h₂lnθ= cst.

(43)

The relaxation method without source term

Initial system:

∂tw+∂xf(w) = 0. (1) Relaxation system:

∂_tW +∂_xF(W) = 1

εR(W), (2)

◮ (2) should formally gives back (1) whenε→0.

◮ (2) should be “simpler” than (1) (e.g. only linearly degenerate fields)

The relaxation scheme is based on a splitting strategy:

Time evolution: We evolve the initial data by the Godunov scheme for the system∂tW +∂xF(W) = 0 (i.e. ε= +∞).

Relaxation: We take into account the relaxation source term by solving∂tW = ¹_εR(W) then taking the limit forε→0.

(44)

Well-balanced relaxation schemes Relaxation models

The Suliciu model

([Suliciu ’98], [Bouchut ’04]...)











∂th+∂xhu= 0

∂thu+∂x(hu²+π) =−ghθ∂xZ

∂_thθ+∂_xhθu= 0

∂thπ+∂x(u(hπ+ν²)) = ^h_ε(gh²θ/2−π)

∂_tZ = 0

Eigenvalues Riemann invariants u±^ν_h u±_h^ν, π∓νu, θ, Z

u (×2) u, π, Z

0 hu, π+^ν_h², θ, gθZ +^u₂² −_2h^ν²2

Difficulties to compute the solution of the Riemann problem:

The order of the eigenvalues is not determined a priori.

There are strong nonlinearities in the Riemann invariants for the eigenvalue 0.