A collection of well-balanced finite volume schemes

Christophe BERTHON jointed work with • Ana¨ıs CRESTETTO, University of Nantes (France) • Vivien DESVEAUX, University of Nantes (France) • Fran¸coise FOUCHER, University of Nantes (France) • Christian KLINGENBERG, University of W¨ urzburg (Germany) • Victor MICHEL-DANSAC, University of Nantes (France) • Markus ZENK, University of W¨ urzburg (Germany) SHARK-FV 2017, May 2017

Main motivations □ Hyperbolic systems of conservation law with source term ∂t w + ∂x f (w) = S(w)

w∈Ω

• Steady states ∂x f (w) = S(w)

⇐⇒

Manifold given by M = {w ∈ Ω; g(w) = 0}

□ Finite volume schemes • Numerical approximations of the weak solutions • Approximation / Exact capture of (a part of) M • Robustness • Entropy stability

Outline □ Ripa model - Euler with gravity Non-explicit steady state capturing scheme • Model and main properties • Incomplete relaxation technique • Numerical scheme and main properties □ A chemotaxis model - Shallow-water model with Manning friction Godunov type well-balanced scheme • Godunov type scheme • Characterization of the approximate Riemann solver • Robustness and well-balanced properties □ Shallow-water model with Manning friction: Well-balanced scheme • Godunov type scheme • Characterization of the approximate Riemann solver • Robustness and well-balanced properties

Ripa model (Chertock et al 2013) □ Shallow-water with potential temperature filed  ∂t h + ∂x hu = 0    ( )  2 h { } 3 ∂t hu + ∂x hu2 + gΘ = −gΘh∂x z Ω = w ∈ R ; h > 0, u ∈ R, Θ > 0  2    ∂t hΘ + ∂x hΘu = 0 □ Steady states

Moving steady states u ̸= 0   ∂x hu = 0  hu = cst    )   (   h2 2 Θ = cst ∂x hu + gΘ = −gΘh∂x z  2    2   u   ∂x hΘu = 0 + gΘ(h + z) = cst 2 With u ̸= 0, we recover the general steady states with a gravity constant gΘ • Steady states at rest  u = 0 2  ∂ Θ h + hΘ∂ z = 0 x x 2

unsolvable

• Particular steady states   u=0 u=0       z = cst Θ = cst      h2 Θ = cst  h + z = cst

 u=0     h = cst     z + h ln Θ = cst 2

□ Objectives • Robust finite volume method • Exact capture of the 3 particular steady states at rest • Exact/Approximated solutions of the unsolvable PDE for steady states

Entropic reformulation □ Main properties The Ripa model is hyperbolic The smooth solutions satisfy the additional conservation law ∂t (η(w) + gΘhz) + ∂x (G(w) + gΘhzu) = 0 where we have set u2 h2 η(w) = h + gΘ 2 2 ( 2 ) u G(w) = h + gΘh2 u 2

Partial entropy Partial entropy flux

Unfortunately, w 7→ η(w) is never convex nor concave. It seems the model does not admit entropy pair

□ Equivalent reformulation for weak solutions: Θ = φ(θ)  ∂t h + ∂x hu = 0    ) (  2 h ∂t hu + ∂x hu2 + gφ(θ) = −gφ(θ)h∂x z  2    ∂t hθ + ∂x hθu = 0 □ Entropy inequalities: The smooth solutions of the new reformulation satisfy ) ( ˜ ∂t (˜ η (w) + gφ(θ)hz) + ∂x G(w) + gφ(θ)hzu = 0 u2 h2 η˜(w) = h + gφ(θ) 2 2 ) ( 2 u ˜ G(w) = h + gφ(θ)h2 u 2 The partial entropy function w 7→ η˜(w) is convex iff 1 ′ 2 φ (θ)φ(θ) − φ (θ) > 0 2 ′′

Example

φ(θ) = eθ

and

θ2 ′′ φ(θ) − θφ (θ) + φ (θ) > 0 2 ′

Incomplete relaxation model □ Suliciu model (Bouchut)  ∂t h + ∂x hu = 0    ( 2 )   ∂t hu + ∂x hu + π + gφ(θ)h∂x z = 0      ∂ hθ + ∂ hθu = 0 t x ( )  2  h h  2  ∂ hπ + ∂ (hπ + a )u = gφ(θ) − π  t x   ε 2    ∂t z = 0

z := z(x) ⇒ ∂t z = 0 Equilibrium ε → 0 h2 π = gφ(θ) 2

□ Suliciu type model (Cargo-LeRoux 94, Chalons et al 10)  ∂t h + ∂x hu = 0    ) ( 2    Equilibrium ε → 0 ∂t hu + ∂x hu + π + gφ(θ)h∂x Z = 0       ∂t hθ + ∂x hθu = 0, h2 π = gφ(θ) ( ) 2  h h2  2  ∂t hπ + ∂x (hπ + a )u = gφ(θ) − π  Z=z  ε 2        ∂t hZ + ∂x hZu = h (z − Z) ε

□ Main properties The Suliciu type relaxation model is hyperbolic with eigenvalues u ± a/h (simple) and u (triple). All fields are linearly degenerated. The eigenvalue u admits only one Riemann invariant (instead of 2). The Riemann problem is thus illposed. □ Additional arbitrary relation To mimic the unsolvable PDE h2 ∂x gΘ = −ghΘ∂x z 2

⇐⇒ equilibrium

uL −

a hL

uR + a hR ⋆ wL

wL

∂x π = −ghΘ∂x z

We propose ⋆ ⋆ ¯ L , WR ))h(W ¯ L , WR )(Z ⋆ − Z ⋆ ) πR − πL = −gφ(θ(W R L

¯ and θ¯ are suitable averages where h

u⋆

⋆ wR

wR

□ Suliciu approximate Riemann solver W = t (h, hu, hθ, hπ, hZ) ( ( ) ) 2 h W eq (w) = t h, hu, hθ, h gφ(θ) , hz 2 ) (x ; WL , WR exact Riemann solution for the relaxation model WR t Theorem With the additional relation, there exists a unique Riemann solution. Moreover ( ) (h,hu,hθ) x WR ; W eq (wL ), W eq (wR ) t defines an approximate Riemann solver (in the sense of Harten, Lax and van Leer) for the Ripa model

Local well-balanced property • Assume wL and wR be given by the steady state relation (general)   uL = uR = 0 2 2 h h  φ(θ ) R − φ(θ ) L + φ(θ(w ¯ L , wR ))h(w ¯ L , wR )(zr − zL ) = 0 R L 2 2 then the approximate Riemann solver stays at rest { (x ) wL (h,hu,hθ) eq eq ; W (wL ), W (wR ) = WR t wR

if x < 0 if x > 0

• Assume wL and wR be given by the steady state relation (particular 1)  uL = uR = 0  1   ¯ h(wL , wR ) = (hL + hR ) 2 θL = θR where   free from θ¯  hL + zL = hR + zR then the approximate Riemann solver stays at rest

• Assume wL and wR be given by the steady state relation (particular 2)  uL = uR = 0  1   ¯ h(wL , wR ) = (hL + hR ) zL = zR 2 where   free from θ¯  φ(θ )h2 = φ(θ )h2 L

L

R

R

then the approximate Riemann solver stays at rest • Assume wL and wR be given by the steady state relation (particular 3)  uL = uR = 0     hL = hR     zL + hL ln φ(θL ) = zR + hR ln φ(θR ) 2 2

where

¯ L , wR ) = 1 (hL + hR ) h(w 2  ( ) φ(θ ) − φ(θ )  R L  φ−1 ¯ L , wR ) = ln(φ(θR )) − ln(φ(θL )) θ(w  θ L

then the approximate Riemann solver stays at rest

if θL ̸= θR , if θL = θR

Relaxation scheme □ Numerical approximation: (win )i∈Z known at time tn (h,hu,hθ) x n , wn ) W ( ; wi−1 i R t

CFL restriction ai±1/2 1 ∆t n max ui ± ≤ n ∆x i∈Z hi 2 win+1 =

1 ∆x

∫

n wi−1

(h,hu,hθ)

xi− 1 2

1 ∆x

WR ∫

xi+ 1 2

xi

x − xi− 12 ∆t (

(h,hu,hθ)

WR

(h,hu,hθ) x n , wn ) ( ; wi i+1 R t

n wi x i− 1 2

(

xi

W

xi

n wi+1 x i+ 1 2

) n ; W eq (wi−1 ), W eq (win ) +

x − xi+ 21 ∆t

) n ) ; W eq (win ), W eq (wi+1

Theorem • The scheme is robust: win+1 ∈ Ω • The scheme exactly preserves the particular steady states: i.e. if wi0 satisfies  0  0  0 ui = 0  u = 0 u = 0    i      i  0 hi = H zi = Z θi0 = θ or or    0     0  (h0 )2 φ(θ0 ) = P  h  hi + zi = H i i zi + i ln(φ(θi0 )) = P 2 then win = win+1 • If the initial data wi0 is an approximation of the unsolvable steady state PDE as follows: ( ) 0 2 0 2 (hi+1 ) 1 1 0 (hi ) 0 0 0 0 0 ¯ ¯ − φ(θi ) (zi+1 −zi ) = 0 φ(θi+1 ) +φ(θ(wi , wi+1 ))h(wi , wi+1 ) ∆x 2 2 ∆x then the scheme exactly captures this approximation win+1 = win

Numerical experiments (Chertock, Kurganov, Liu 13) □ Perturbation of a nonlinear steady state z(x) = 6 − 2 exp(x)

and

(hs , us , Θs )(x) = (exp(x), 0, exp(2x))

0.04

Initial perturbation t=0.2 t=0.4

0.1

t=0.2 t=0.4

0.03 0.02

0.08

0.01 u−us

h−hs

0.06

0.04

0 −0.01 −0.02

0.02

−0.03 −0.04

0 −1

−0.5

0

0.5

1

−0.05 −1

−0.5

0

0.5

1

Figure 1: Perturbation of the free surface h − hs (left) and perturbation of the velocity u − us (right)

□ Dam break over a non-flat bottom 5 5 4.5

4

4

3 θ

h+z

3.5

2

3 2.5 2

1

1.5 1

0 −1

−0.5

12 10

p

8 6 4 2 0

0

0.5

1

−1

−0.5

0

0.5

1

Free surface h + z (top left) Temperature Θ (top right) Pressure p (bottom) at time t = 0.2

Euler with gravity (Xu et al 2010-2011, K¨ appeli and Mishra 2014)  ∂t ρ + ∂x ρu = 0    ∂t ρu + ∂x (ρu2 + p(ρ, e)) = −ρ∂x Φ    ∂ E + ∂ (E + p(ρ, e))u = −ρu∂ Φ t

x

u2 E = ρe + ρ 2 Φ gravity field Ω = {w ∈ R3 ; ρ > 0, u ∈ R, e > 0}

x

□ Entropy inequalities ∂t ρF(s) + ∂x ρF(s)u ≤ 0 □ Steady states at rest { u=0 ∂x p = −ρ∂x Φ

(Gibbs law)

Isothermal steady states u=0

ρ = αe

−βΦ

α −βΦ p= e β

□ Objectives • Robust and stable finite volume scheme • Exact capture of the isothermal steady states • Exact/Approximated solutions of the unsolvable PDE for steady states

Incomplete relaxation model □ Suliciu type model (Cargo-LeRoux 94, Chalons et al 10)  ∂t ρ + ∂x ρu = 0    ( 2 )   Equilibrium ε → 0  ∂t ρu + ∂x ρu + π = −ρ∂x Z     ∂t E + ∂x (E + π) u = −ρu∂x Z π = p(ρ, e)  ρ  2 Z=Φ  ∂ ρπ + ∂ (ρπ + a )u = (p(ρ, e) − π)  t x   ε     ∂ ρZ + ∂ ρZu = ρ (Φ − Z) t x ε □ Additional closure relation ⋆ ⋆ ⋆ πR − πL = −¯ ρ(ρL , ρR )(ZR − ZL⋆ )

to mimic ∂x p = −ρ∂x Φ

Definition of an approximate Riemann solver ) ( (h,hu,hθ) x eq eq WR ; W (wL ), W (wR ) t with good properties: Robustness - Local well-balance - Stability

Relaxation scheme

win+1 =

1 ∆x

∫

(

xi

(h,hu,hθ)

xi− 1 2

1 ∆x

WR ∫

xi+ 1 2

xi

x − xi− 12 ∆t (

(h,hu,hθ)

WR

) n ; W eq (wi−1 ), W eq (win ) +

x − xi+ 21 ∆t

) n ; W eq (win ), W eq (wi+1 )

Theorem With a relaxation parameter ai+1/2 >> 1 and under a relevant CFL restriction: • Robustness: win+1 ∈ Ω • Entropy preserving: n+1 ρn+1 F(s ) i i

≤

ρni F(sni )

) ∆t ( n n − {ρF(s)u}i+1/2 − {ρF(s)u}i+1/2 ∆x

• Steady state preserving: With an initial data given by 1 0 Φi+1 − Φi (pi+1 − p0i ) + ρ(ρ0i , ρ0i+1 ) =0 ∆x ∆x Then win+1 = win

• Hydrostatic steady state preserving: Assume ρ=

ρR − ρL ln(ρR ) − ln(ρR )

and an initial data given by the hydrostatic atmosphere u0i Then win+1 = win

=0

ρ0i

= αe

−βΦi

p0i

α −βΦi = e β

Numerical Results: Small perturbation of an hydrostatic atmosphere • Computational domain: [0, 1] until time T = 0.25 • Φ(x) = x • Initial condition:

 −x  ρ (x) = e 0   u0 (x) = 0    −x 100(x−0.5)2 p0 (x) = e + 0.01e

• Pressure perturbation: δp = p − p0

0.01 Initial perturbation Reference solution Solution with 100 cells

pressure perturbation

0.008

0.006

0.004

0.002

0

Numerical Results: Non-hydrostatic steady state • Computational domain: [0, 1] with periodic boundary conditions • Final time T = 1 • Φ(x) = sin(2πx)

   ρ0 (x) = 3 + 2 sin(2πx) • Initial condition: u0 (x) = 0    p0 (x) = 3 + 3 sin(2πx) − To have ∂x p0 + ρ0 ∂x Φ = 0 N

• L1 error:

Density

1 2

cos(4πx)

Velocity

100

4.46E-05

–

2.03E-05

–

200

7.11E-06

2.65

5.29E-06

1.94

400

1.23E-06

2.53

1.34E-06

1.98

800

2.35E-07

2.39

3.37E-07

1.99

1600

5.02E-08

2.23

8.44E-08

2.00

3200

1.15E-08

2.13

2.11E-08

2.00

A model of Chemotaxis (Ribot et al 2012 - 2014)  ∂t ρ + ∂x ρu = 0    ∂t ρu + ∂x (ρu2 + p(ρ)) = −χρ∂x Φ − αρu    ∂ Φ − D∂ Φ = aρ − bΦ t

χ, α, D, a, b given parameters Ω = {w ∈ R3 ; ρ ≥ 0, u ∈ R, Φ ≥ 0} p(ρ) = δργ

xx

□ Steady states at rest  u = 0  e(ρ) − χΦ = cste

e(ρ) = δ

γ ργ−1 + cste γ−1

solutions of D∂xx Φ − bΦ = aρ if ρ = 0 : if ρ > 0, C < 0 : if ρ > 0, C > 0 :

( √ ) ( √ ) ϕ (x) = A cosh x b + B sinh x b , ( √ ) ( √ ) ϕ (x) = A cos x |C| + B sin x |C| − ϕp , ( √ ) ( √ ) ϕ (x) = A cosh x C + B sinh x C − ϕp ,

where A and B are some constants, C =

1 D

( b−

aχ 2ε

)

ρ (x) = ρ (x) =

and ϕp =

χ 2ε χ 2ε

(ϕ (x) − K) , (ϕ (x) − K) ,

Kaχ 2εb−aχ .

□ Asymptotic behavior Rescaling: t → t/ε (long time) and α → α/ε (dominant friction) Limit ε → 0 to get a diffusive regime { ∂t ρ = ∂x (∂x p − χρ∂x Φ) D∂xx Φ = bΦ − aρ □ Objectives • Robustness (ρ ≥ 0 and Φ ≥ 0) • Steady state preserving (well-balanced) • Asymptotic preserving Godunov type strategy (CB and Chalons 16)

Godunov type scheme {

∂t ρ + ∂x ρu = 0 ∂t ρu + ∂x (ρu2 + p(ρ)) = −χρ∂x Φ − αρu

Φ given

□ Approximate Riemann solver: w ˜ λL

0 ⋆ wL

λR ⋆ wR

wL

λL < 0 < λR HLL type solver Source term → stationary contact wave

wR

x

• Harten-Lax-van Leer consistency condition ∫ ∆x/2 ∫ ∆x/2 1 1 w ˜ (x, ∆t; wL , wR ) dx = wR (x, ∆t; wL , wR ) dx ∆x −∆x/2 ∆x −∆x/2 1 ∆x

∫

∆x/2

1 ∆t ⋆ ⋆ w ˜ (x, ∆t; wL , wR ) dx = (wL +wR )+ (λL (wL −wL )+λR (wR −wR )) 2 ∆x −∆x/2

Because tof the source term, wR stays unknown wR (x, t; wL , wR ) wR (t)

wL (t) wL

1 ∆x

wR

∫

x

constant is not a natural solution ∫ ∆x/2 ∫ ∆t ( ) ∂t w + ∂x f (w) = S(w) dxdt −∆x/2

0

∫ ∆x/2 ∫ ∆t 1 1 wR (x, ∆t; wL , wR ) dx = (wL + wR ) + S(w)dxdt 2 ∆x −∆x/2 −∆x/2 0 ∫ ∆t ∫ ∆t 1 1 − f (wR (∆x/2, t))dt + f (wR (−∆x/2, t))dt ∆x 0 ∆x 0 ∆x/2

Approximation wR (−∆x/2, t) ≃ wL wR (∆x/2, t) ≃ wR

As a consequence ∫ ∆x/2 ∆t 1 1 ρR (x, ∆t; wL , wR ) dx ≃ (ρL + ρR ) − (ρR uR − ρL uL ) ∆x −∆x/2 2 ∆x ∫ ∆x/2 1 1 ∆t (ρu)R (x, ∆t; wL , wR ) dx ≃ (ρL uL + ρR uR ) − (ρR u2R + pR − ρL u2L − pL ) ∆x −∆x/2 2 ∆x ∫ ∆t ∫ ∆x/2 1 + ∆tSR − α (ρu)R (x, t; wL , wR ) dxdt ∆x 0 −∆x/2 ∫ ∆t ∫ ∆x/2 1 SR = χρR ∂x Φdxdt ∆t∆x 0 −∆x/2 • Approximation SR ≃ S ⋆ to be defined (independently from ∆t) • Approximation ∫ ∆x/2 1 (ρu)R (x, ∆t; wL , wR ) dx ≃ F (∆t) ∆x −∆x/2 solution of the following integral equation 1 ∆t F(∆t) = (ρL uL + ρR uR ) − (ρR u2R + pR − ρL u2L − pL ) + ∆tS ⋆ − α 2 ∆x

∫

∆t

F(t)dt 0

□ Consistency conditions λL (ρL − ρ⋆L ) + λR (ρ⋆R − ρR ) = ρL uL − ρR uR λL (ρL uL − ρ⋆L u⋆L ) + λR (ρ⋆R u⋆R − ρR uR ) = ) ) (α 1 ( −α∆t e −1 (ρL uL + ρR uR )∆x − (ρR u2R + pR − ρL u2L − pL ) + ∆xS ⋆ α∆t 2 □ Flux continuity: ρ⋆L u⋆L = ρ⋆R u⋆R □ Well-balanced conditions χ pR − pL (ΦR − ΦL ) S = ∆x eR − eL ρ⋆L ρ⋆R eL − χϕL = eR − χϕR ρL ρR ⋆

□ Approximate Riemann solver (Munz et al 09): w ˜ λL

0 Φ⋆ L

λR

{

Φ⋆ R

ΦL

ΦR

∂t Φ + ∂x Ψ = aρ − bΦ Ψ = ∂x Φ

ρ given

x

□ Harten-Lax-van Leer consistency condition ∫ ∆x/2 ∫ ∆x/2 1 ˜ (x, ∆t; wL , wR ) dx = 1 Φ ΦR (x, ∆t; wL , wR ) dx ∆x −∆x/2 ∆x −∆x/2 Main difficulty: Evaluate ΦR ∫ ∆x/2 ∫ ∆t ∫ (∂t Φ − D∂x Ψ) dxdt = a −∆x/2

0

∆x/2 −∆x/2

∫

∫ ∫

∆t

ρR (x, t)dxdt−b 0

Approximations ∫ ∆x/2 t 1 1 (ρR uR − ρL uL ) ρR (x, t)dx ≃ (ρL + ρR ) − ∆x −∆x/2 2 ∆x ∫ ∆x/2 ∫ ∆t 1 ∆t ∂x Ψ(x, t)dx ≃ (ΨR − ΨL ) ∆x −∆x/2 0 ∆x

ΦR (x, t)dxdt

Godunov type scheme □ Approximate Riemann solver: w ˜ λL

0 Φ⋆ L

λR

{

Φ⋆ R

ΦL

ΦR

∂t Φ + ∂x Ψ = aρ − bΦ Ψ = ∂x Φ

ρ given

x

□ Harten-Lax-van Leer consistency condition ∫ ∆x/2 1 ΦR (x, t)dx ≃ G(∆t) ∆x −∆x/2 G(∆t) solution of the following integral equation ( ) 2 ∆t ∆t ∆t 1 (ΨR − ΨL ) + a (ρL + ρR ) − (ρR uR − ρL uL ) G(∆t) = (ΦL + ΦR ) + D 2 ∆x 2 2∆x ∫ ∆t G(t)dt −b 0

□ Godunov type scheme  ( ( ) ( )) n+1 ∆t ⋆ ⋆  w = win − ∆x λi− 21 ,R win − wi− − λi+ 12 ,L win − wi+ 1 1 i ,R 2 2 ,L)) ( ) ( ( n ⋆ n ⋆  Φn+1 = Φn − ∆t λ 1 1 Φ − λ Φ − Φ − Φ 1 i− ,R i+ ,L i i i i ∆x i− ,R i+ 1 ,L 2 2 2

2

□ Definition of Ψni 1 (Φni+1 − Φni−1 ) × E(∆x) 2∆x E(∆x) is consitent with 1 lim E(∆x) = 1 Ψni =

∆x→0

We impose

 b ∆x2   ( )  √  2 cos  b∆x − 1        ∆x2 C ( ) √ E(∆x) = 2 cos  |C|∆x − 1      2  ∆x C   ( )  √   2 cosh C∆x − 1

to recover the steady states

if ρ = 0

if ρ > 0, C < 0

if ρ > 0, C > 0

Theorem • Robustness (adopting a local cut-off, Chalons 2014) ρ⋆L,R = min(max(0, ρ⋆L,R ), 2ρHLL ) Φ⋆L,R = min(max(0, Φ⋆L,R ), 2ΦHLL ) • Well-balance property • Asymptotic preserving

⇒

ρn+1 ≥0 i

and

Φn+1 ≥0 i

Numerical results Influence of γ at χ = 10, L = 3, ∆x = 0.01. 8

30

=3

=2 7 25

6 20

5

4

15

3 10

2 5

1

0

0 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

x

x

1.8

3.5

=5

=4 1.6 3

1.4 2.5

1.2 2

1

0.8

1.5

0.6 1

0.4 0.5

0.2

0

0 0

0.5

1

1.5

x

2

2.5

3

0

0.5

1

1.5

x

2

2.5

3

Shallow-water model with Manning friction (Vazquez-Cend´ on)    ∂t h + ∂x hu = 0 ( ) 2 h gκ 2  = − η |hu|hu − gh∂x z  ∂t hu + ∂x hu + g 2 h ∂t w + ∂x f (w) = S(w) Ω = {w ∈ R2 ; h > 0} To simplify, we fix z = cst ⇒ □ Steady states

gκ S = − η |hu|hu h

   ∂x hu = 0 ) ( 2 gκ h  = − η |hu|hu  ∂x hu2 + g 2 h

With hu = 0, we recover the usual shallow-water steady states

η>2

□ Steady states    hu = cst 2 g (h u ) 0 0 η+2  (hη+2 − h0 ) − (hη−1 − h0η−1 ) = −gκ|h0 u0 |h0 u0 (x − x0 )  η+2 η−1 where h0 = h(x0 ) and u0 = u(x0 ) Remark • Steady states are not globally defined (∀x ∈ R) □ Objectives: Numerical scheme preserving the main properties • Steady states preserving (well-balanced) • Water height positive preserving

Approximate Riemann solver □ Two intermediate constant states λL

0 ⋆ wL

wL

λR ⋆ wR wR

x

λL < 0 < λR HLLC type solver Source term → stationary contact wave 4 unknowns: h⋆L,R and u⋆L,R

• Consistency condition ∫ ∆x ∫ ∆x 2 2 1 x 1 x w( ˜ ; wL , wR ) dx = wR ( ; wL , wR ) dx ∆x − ∆x t ∆x − ∆x t 2 2

Approximate Riemann solver □ Two intermediate constant states λL

0 ⋆ wL

λR ⋆ wR

wL

wR

x

λL < 0 < λR HLLC type solver Source term → stationary contact wave 4 unknowns: h⋆L,R and u⋆L,R

• Consistency condition  ⋆ ⋆ h u − h u = λ (h − h ) + λ (h L L R R L L R L R − hR )    ) ( ) ( 2 2 h h hL u2L + g L − hR u2R + g R + ∆x S¯LR =  2 2    λL (hL uL − h⋆L u⋆L ) + λR (h⋆R u⋆R − hR uR ) where We approximate S¯LR

1 S¯LR = ∆t∆x ,→

∫

∆x 2

− ∆x 2

∫

∆t

− 0

gκ η |hR uR |hR uR dx dt hR

We get an additional unknown

• Contact preserving: h⋆L u⋆L = h⋆R u⋆R = q ⋆ • Steady state preserving: S¯LR is defined by enforcing

   hL uL = hR uR = q0 [ ] 2 g q0 η−1 η+2  h − h  − gκ|q0 |q0 ∆x = η+2 η−1 LR to get

[ S¯LR = −gκ|q ⋆ |q ⋆ [

(q ⋆ )2 h

g η+2 h η+2

{ and

2

+ g h2

−

h⋆L = hL

u⋆L = uL

h⋆R = hR

u⋆R = uR

] LR

(q ⋆ )2 η−1 η−1 h

] LR

,→ It defines a consistent average ,→ It is a necessary condition to recover all steady states • Additional relation (contact preserving): ) ( ) ( ⋆ 2 ⋆ 2 (q ) g (q ) g (h⋆R )η+2 − (h⋆R )η−1 − (h⋆L )η+2 − (h⋆L )η−1 −gκ|q ⋆ |q ⋆ ∆x = η+2 η−1 η+2 η−1 Remark Nonlinear equations are solved by a Newton method

Numerical experiments □ Perturbation of a nonlinear steady state

Figure 2: Perturbation of the free surface h

Numerical experiments □ Dam break

Thanks for your attention