the rs-imex scheme for the low-froude shallow water equationsmhvignal/exposes/... · 2017. 11....

The RS-IMEX scheme forthe low-Froude shallow water equations¶

Hamed Zakerzadeh

joint work with: Sebastian Noelle

Institut de Mathématiques de Toulouse, Université Toulouse III - Paul Sabatier

Toulouse, FranceNov. 22nd 2017

¶This research was supported by the scholarship of RWTH Aachen university through Graduiertenförderung nach Richtlinien zurFörderung des wissenschaftlichen Nachwuchses (RFwN).

Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent progress

Outline

Introduction

RS-IMEX scheme

1d SWE

2d SWENumerical experiments

2d RSWENumerical experiments

Recent progress

1/34


2/34


I g : gravity acceleration

I f : Coriolis parameter

I p : pressure

I x := (x1, x2, x3)T

I u := (u1, u2, u3)T

Compressible Euler Equations∂t%+ divx (%u) = 0

∂t(%u) + divx (%u ⊗ u + pI3) =

Gravitation︷︸︸︷−%g k̂

Coriolis︷︸︸︷−%f k̂ × u

3/34


I homogeneity =⇒ % constant

I incompressibility =⇒ divxu = 0I shallowness =⇒ ∂x3p = −%g =⇒ u3 ∼ O(δ)

Lh ∼ 102–103 kmLv ∼ 1–5 kmδ :=

LhLv∼ 10−3 − 10−2

I boundary conditions:I no normal flow at bottomI free surface at top

Rotating Shallow Water Equations in horizontal plane (x1, x2){∂th + divx (hu) = 0

∂t(hu) + divx(hu ⊗ u + gh

2

2I2)

= −gh∇xηb − fhu⊥

ηb is the bottom function, u⊥ := (−u2, u1).

4/34


I homogeneity =⇒ % constantI incompressibility =⇒ divxu = 0

I shallowness =⇒ ∂x3p = −%g =⇒ u3 ∼ O(δ)

Lh ∼ 102–103 kmLv ∼ 1–5 kmδ :=

LhLv∼ 10−3 − 10−2




2

2I2)



4/34


I homogeneity =⇒ % constantI incompressibility =⇒ divxu = 0I shallowness =⇒ ∂x3p = −%g =⇒ u3 ∼ O(δ)

Lh ∼ 102–103 kmLv ∼ 1–5 kmδ :=

LhLv∼ 10−3 − 10−2




2

2I2)



4/34


Non-dimensionalisation

x̂ :=xL◦, t̂ :=

t

t◦, û :=

uu◦, ĥ :=

h

H◦, η̂b :=

ηb

H◦, t◦ =

L◦

u◦

Non-dimensionalised RSWESt∂t̂ ĥ + divx̂ (ĥû) = 0

St∂t̂(ĥû) + divx̂

(ĥû ⊗ û +

ĥ2

2Fr2I2

)= −

ĥ

Fr2∇x̂ η̂b −

ĥ

Roû⊥

St :=L◦/u◦

t◦= 1, Fr :=

u◦√gH◦

, Ro :=u◦

L◦f=

f −1

t◦

We consider two singular limits:

I Non-rotating: f = 0 and Fr = ε� 1

I Rotating: Fr ∼ Ro = ε� 1

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Analytical/Numerical issues

What happens if ε→ 0?

I Continuous level: Does the limε→0

system exist?

I comp. Euler → incomp. Euler [Klainerman and Majda, 1981]

I RSWEquasi-geostrophic−−−−−−−−−→distinguished limit

quasi-geostrophic equations [Majda, 2003]

I Discrete/numerical level: How does the scheme behave?

I Stiffness: wave speeds ∼ O( 1ε )- Explicit: CFL condition ∆t . ε∆x

- Implicit: diffuses slow material waves

I Inconsistency with the limit system

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Asymptotic Preserving (AP) schemes

Introduced by [Jin, 1999]

- [Il’in, 1969]

- [Larsen et al., 1987]

I Asymptotic Efficiency (AEf): uniform CFL, efficient implicit step

I Asymptotic Consistency (AC): consistent with the asymptotic system as ε→ 0

I Asymptotic Stability (AS): uniformly stable in ε

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Outline

Introduction

RS-IMEX scheme

1d SWE



Recent progress

7/34


Reference Solution IMEX scheme (I)

Hyperbolic system of balance laws for U ∈ Rq in Ω ⊂ Rd

∂tU(t, x ; ε) + divxF (U, t, x ; ε) = S(U, t, x ; ε),

I IMEX: stiff + non-stiffimplicit + explicit

I How to decompose?I non-linear stiff part → non-linear iteration [Degond and Tang, 2011]I linearly-implicit methods!

∂tU = N (U) =⇒ ∂tU = L(U) + (N −L)(U)

I for ODEs [Rosenbrock, 1963]:

x ′(t) = f (x) =⇒ x ′(t) = f ′(x) x(t) +[f (x(t))− f ′(x(t)) x(t)

]I for Euler with gravity [Restelli, 2007]I penalization method [Filbet and Jin, 2010]: ∂t f + v · ∇x f = 1εQ(f )

Q(f ) = P(f ) + Q(f )− P(f ), P(f ) := Q′(M)(f −M)

8/34


Reference Solution IMEX scheme (II)

U = U︸︷︷︸Reference solution

+ D V︸︷︷︸scaled perturbation

with D = diag(εd1 , . . . , εdq ) for U = U(0) + εU(1) + ε2U(2)

F = F (U) +

Linear︷︸︸︷F ′(U)DV +

Nonlinear︷︸︸︷F̂ (U,V ) = D

RS+IM+EX︷︸︸︷(G + G̃ + Ĝ

)S = S(U) + S ′(U)DV + Ŝ(U,V ) = D

(Z + Z̃ + Ẑ

)

∂tV = −T +

=:R̃︷︸︸︷(−divx G̃ + Z̃

)+

=:R̂︷︸︸︷(−divx Ĝ + Ẑ

)with scaled residual of the reference solution

T := D−1∂tU + divxG − Z

9/34


Reference Solution IMEX scheme (II)

U = U︸︷︷︸Reference solution

+ D V︸︷︷︸scaled perturbation

with D = diag(εd1 , . . . , εdq ) for U = U(0) + εU(1) + ε2U(2)

F = F (U) +

Linear︷︸︸︷F ′(U)DV +

Nonlinear︷︸︸︷F̂ (U,V ) = D

RS+IM+EX︷︸︸︷(G + G̃ + Ĝ

)S = S(U) + S ′(U)DV + Ŝ(U,V ) = D

(Z + Z̃ + Ẑ

)

∂tV = −T +

=:R̃︷︸︸︷(−divx G̃ + Z̃

)+

=:R̂︷︸︸︷(−divx Ĝ + Ẑ

)with scaled residual of the reference solution

T := D−1∂tU + divxG − Z9/34


Reference Solution IMEX scheme (III)

RS-IMEX scheme

DtV n∆ = −Tn+1

∆ + R̃n+1∆ + R̂

n∆

I Time integration Dtφ(t, x) := φ(t+∆t,x)−φ(t,x)∆tI Rusanov-type flux fi+1/2 :=

f (ui )+f (ui+1)2

−αi+1/2

2(ui+1 − ui )

I Central discretization of the source term

1: get Un∆ and V n∆

2: find Un+1∆ → compute Tn+1

∆

3: Explicit step DtV n∆ = R̂n∆ → V

n+1/2∆

4: Implicit step DtVn+1/2

∆ = −Tn+1

∆ + R̃n+1∆ → V

n+1∆

5: Un+1∆ = Un+1

∆ + DVn+1

∆

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a bit of literature review

System Reference solution U[Bispen et al., 2014] SWE LaR[Schütz and Kaiser, 2016] Van der Pol ∞ damping[Zakerzadeh, 2016a] 1dSWE LaR and lake eq.[Kaiser et al., 2016] isent. Euler incompressible eq.[Zakerzadeh, 2016b] 2dSWE lake eq.[Bispen et al., 2017] Euler + gravity hydrostatic equilib.[Zakerzadeh, 2017b] 2dSWE + Coriolis barotropic vorticity eq.[Kaiser and Schütz, 2017] extends [Kaiser et al., 2016] to high order dG

I Non-rotating:[Degond and Tang, 2011; Haack et al., 2012; Noelle et al., 2014; Bispen et al., 2014;Dimarco et al., 2016; Zakerzadeh, 2017a], . . .

I Rotating:[Bouchut et al., 2004; Audusse et al., 2009, 2011, 2015, 2017; Lukáčová-Medvid’ováet al., 2007; Hundertmark-Zaušková et al., 2011]

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Outline

Introduction

RS-IMEX scheme

1d SWE



Recent progress

11/34


1d Shallow water equations with topography

∂th + ∂x (hu) = 0∂t(hu) + ∂x (hu2 + h22ε2

)= −

h

ε2∂xηb

Define: [Bispen et al., 2014]

I Hmean − ηb =: −b > 0I h = z − bI m := hu

U =[zm

], F =

mm2z − b

+z2 − 2zb

2ε2

, S = [ 0− zε2∂xb

].

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I U := (z, 0)T lake at rest =⇒ T = 0I D := diag(ε2, 1) ⇐= η(0) = const.

Ĝ =

0v22z + ε2v1 − b

+ε2

2v21

G̃ = [ v2/ε2(z − b)v1

]

Ẑ = 0 Z̃ =[

0−∂xb v1

]λ̂ = 0, 2 upert λ̃ = ±

√z − bε

V n+1/2i = Vni −

∆t

∆x

(Ĝni+1/2 − Ĝ

ni−1/2

)(Explicit step)

V n+1i = Vn+1/2i −

∆t

∆x

(G̃n+1

i+1/2− G̃n+1

i−1/2

)+ ∆t Z̃n+1i (Implicit step)

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Asymptotic analysis

Asymptotic analysis in 1d

Theorem [Zakerzadeh, 2016a]

For 1d SWE with topography in Ω = T and under an ε-uniform CFL condition, withwell-prepared initial data and LaR reference solution, the RS-IMEX scheme is

(i) solvable: a unique solution for all ε > 0

(ii) ε-stable: limε→0‖V n+1∆ ‖ = O(1)

(iii) Rigorously AC: for the fully-discrete settings

(iv) AS: there exists a constant CN,Tf such that

‖V n∆ ‖`2 ≤ CN,Tf ‖V0

∆ ‖`2

(v) well-balanced: preserves the lake at rest equilibrium state

(vi) possible O(ε2) checker-board oscillations for z

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Asymptotic analysis

Sketch of the proof: asymptotic consistency

I recast the linear implicit step as JεV n+1∆ = Vn+1/2∆︸︷︷︸O(1)

Jε :=

[IN

β

ε2Q

βRb IN

], β :=

∆t

2∆x, α̃ = 0

Q := Circ (0, 1, 0, . . . , 0,−1)

(Rb)i = (bi+1 − bi−1, hi+1, 0, . . . , 0,−hi−1)

I show that limε→0‖J−1ε ‖


Asymptotic analysis

Sketch of the proof: asymptotic stability

V n∆ = Eimp EexpVn−1

∆

I Implicit: ‖Eimp‖`2 ≤ 1 + Cimp∆tI Explicit: ‖EexpV n−1∆ ‖`2 ≤ ‖V

n−1∆ ‖`2 + Cexp∆t‖V

n−1∆ ‖

2`2

‖V n∆ ‖`2 ≤ (1 + Cimp∆t)(

1 + Cexp∆t ‖V n−1∆ ‖`2)‖V n−1∆ ‖`2

discrete Gronwall’s inequality [Willett and Wong, 1965]=⇒ requires smallness of the initial datum

16/34


Outline

Introduction

RS-IMEX scheme

1d SWE



Recent progress

16/34


2d Shallow water equations with topography

F =

m1 m2

m21z − b

+z2 − 2zb

2ε2m1m2

z − bm1m2

z − bm22

z − b+

z2 − 2zb2ε2

, S = 0−z∂xb/ε2−z∂yb/ε2

.

G := G(U), G̃ := G ′(U)V , Ĝ := G − G − G̃

Z := Z(U), Z̃ := Z ′(U)V , Ẑ := Z − Z − Z̃

I U : zero-Froude limit (lake equations){divxm = 0∂tm − divx ( m⊗mb )− b∇xπ = 0

I Chorin’s projection method to update UI D = diag(ε2, 1, 1)

17/34


V n+1/2ij = Vnij −

∆t

∆x

(Ĝn1,i+1/2j − Ĝ

n1,i−1/2j

)−

∆t

∆y

(Ĝn2,i j+1/2 − Ĝ

n2,i j−1/2

)V n+1ij = V

n+1/2ij −

∆t

∆x

(G̃n+1

1,i+1/2j− G̃n+1

1,i−1/2j

)−

∆t

∆y

(G̃n+1

2,i j+1/2− G̃n+1

2,i j−1/2

)+ ∆t Z̃n+1ij −∆t T

n+1ij

T n+11,ij =(∇h,xm1n+1ij +∇h,xm2

n+1ij

)/ε2

T n+12,ij = Dtm1nij +∇h,x

(m1

n+1,2ij

z − bij

)+∇h,y

(m1

n+1ij m2

n+1ij

z − bij

)

T n+13,ij = Dtm2nij +∇h,x

(m1

n+1ij m2

nij

z − bij

)+∇h,y

(m1

n+1,2ij

z − bij

)

18/34


Asymptotic analysis

Asymptotic analysis in 2d

Theorem [Zakerzadeh, 2016b]

For 2d SWE with topography in Ω = T2 and under an ε-uniform CFL condition, withwell-prepared initial data and zero-Froude limit reference solution, the RS-IMEXscheme is


(ii) “ε-stable”: limε→0‖V n+1∆ ‖ = O(1) (if U = 0 and ∇xη

b = 0)


(iv) “AS”: there exists a constant CN,Tf such that

‖V n∆ ‖`2 ≤ CN,Tf ‖V0

∆ ‖`2

provided the reference solver is stable in some suitable sense

(v) “well-balanced”: preserves the LaR state (if U∆,V∆ ∈ ULaR∆ )(vi) possible O(ε2) checker-board oscillations for z

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Numerical experiments

Travelling vortex

I Exact solution is available [Ricchiuto and Bollermann, 2009]

I Initial condition as [Bispen et al., 2014] with periodic domain Ω = [0, 1)2:

z(0, x , y) = 1[r≤ π

ω]

(Γε

ω

)2(g(ωr)− g(π)) ,

u1(0, x , y) = u0 + 1[r≤ πω

]Γ (1 + cos(ωr)) (yc − y),u2(0, x , y) = 1[r≤ π

ω]Γ (1 + cos(ωr)) (x − xc ),

with b(x , y) = −110, u0 = 0.6 and

r := dist(x , xc ), xc = (0.5, 0.5)T , Γ = 1.4, ω = 4π,

g(r) := 2 cos r + 2r sin r +1

8cos 2r +

r

4sin 2r +

3

4r2.

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Travelling vortex: initial condition

Initial condition for the travelling vortex example with ε = 0.8, computed on the 100 × 100 grid.

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Uniform accuracy (I)

ε = 0.8. ε = 0.01.

Error of the RS-IMEX scheme, computed on the 80 × 80 grid with CFL = 0.45 and Tf = 1

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Uniform accuracy (II)

Experimental order of convergence for the travelling vortex example with Tf = 1.

ε = 0.8N ez,`∞ EOCz,`∞ eu1,`∞ EOCu1,`∞20 2.61e-2 - 1.04e-1 -

40 2.00e-2 0.38 6.80e-2 0.61

80 1.23e-2 0.70 3.63e-2 0.91

160 6.20e-3 0.99 1.65e-3 1.14

ε = 10−6

N ez,`∞ EOCz,`∞ eu1,`∞ EOCu1,`∞20 4.08e-14 - 1.04e-1 -

40 3.13e-14 0.38 6.80e-2 0.61

80 1.92e-14 0.71 3.63e-2 0.91

160 9.69e-15 0.99 1.65e-3 1.14

23/34



Should we invest in U?

I solving for U takes around 1% of the total timeI the quality matters!

ε = 0.01, U = U(0) . ε = 0.01, U = 0 .

24/34


Outline

Introduction

RS-IMEX scheme

1d SWE



Recent progress

24/34


2d Rotating shallow water equations with topography

∂t̂(Θẑ) + divx̂ (ĥû) = 0,

∂t̂(ĥû) + divx̂

(ĥû ⊗ û +

ĥ2

2Fr2I2

)= −

Θ

Fr2ĥ∇x̂ η̂b −

ĥ

Rou⊥,

I two height scales: H◦ for Hmean, and Z◦ for z and ηb

I ĥ = 1 + Θ(ẑ − η̂b)I Θ := Z◦

H◦

I F 1/2 := fL◦/√gH◦ = O(1)

Quasi-geostrophic distinguished limit [Majda, 2003]

Ro = ε� 1, Fr = F 1/2ε, Θ = Fε

Θ ∼ ε =⇒ the variation of ηb and z are mild:

‖z‖, ‖∇xηb‖ = O(ε)

25/34


∂tz +1Θdivxm = 0,

∂tm + divx(

m ⊗mΘz − b

+Θz2 − 2bz

2εI2)

= −1

εz∇xb −

1

εm⊥,

with m := (Θz − b)u and 1−Θηb = −b.

F =

m1/Θ m2/Θ

m21Θz − b

+Θz2 − 2zb

2ε

m1m2

Θz − bm1m2

Θz − bm22

Θz − b+

Θz2 − 2zb2ε

SB =

0−z∂xb/ε−z∂yb/ε

SC = 0m2/ε−m1/ε

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I U : quasi-geostrophic equations [Majda, 2003]

u = ∇⊥x z (geostrophic balance)∆xz = ζ, with ζ := ‖∇x × u‖

(∂t + u · ∇x ) (ζ − Fz + Fηb) = 0 (potential vorticity eq.),

I Arakawa method to update U [Arakawa, 1966]I D = I3

G := G(U), G̃ := G ′(U)V , Ĝ := G − G − G̃

Z := Z(U), Z̃ := Z ′(U)V , Ẑ := Z − Z − Z̃

27/34


Asymptotic analysis

Asymptotic analysis in 2d rotating case

Theorem [Zakerzadeh, 2017b]

For 2d RSWE with topography in Ω = T2 and under an ε-uniform CFL condition, withwell-prepared initial data and the QGE reference solution, the RS-IMEX scheme is


(ii) “ε-stable”: limε→0‖V n+1∆ ‖ = O(1)


(iv) “AS”: there exists a constant CN,Tf such that

‖V n∆ ‖`2 ≤ CN,Tf ‖V0

∆ ‖`2

provided the reference solver is stable in some suitable sense

(v) “well-balanced”: preserves the LaR state if U∆,V∆ ∈ ULaR∆(vi) possible O(ε) checker-board oscillations for the surface perturbation

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2d stationary vortex

in periodic domain [0, 1)2, pressure gradient is balanced with the Coriolis force and theadvective terms [Audusse et al., 2009]:

u0(r , θ) = ϑθ(r)θ̂, ϑθ(r) := 5r1[r< 15

] + (2− 5r)1[ 15≤r< 2

5],

z ′0(r) = ϑθ + εϑ2θr,

where r is the distance to the vortex center (0.5, 0.5)T and Hmean = 2.

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Relative error for z.|z∆(t) − z∆(0)| for t = 10 and ε = 10

−4.

=⇒ the scheme is uniformly accurate and AS!

30/34



Absolute divergence of the velocity field for ε = 10−4 at t = 1. Geostrophic balance for ε = 10−4 at t = 1.

=⇒ the scheme is AC!

31/34



Should we invest in U?

ε = 0.1, U = U(0) . ε = 0.1, U = 0 .

32/34


Outline

Introduction

RS-IMEX scheme

1d SWE



Recent progress

32/34


How to refine estimates?I following [Giesselmann, 2015]: (semi-discrete)

I for Etot :=12‖h|u|

2‖L1(Ω) +1

2ε2‖z‖2L2(Ω) and all ε > 0

E n+1tot ≤ Entot +O(∆t

2)

I for Ekin,(0) :=12‖|u(0)|

2‖L1(Ω), when ε→ 0

E n+1kin,(0) ≤ Enkin,(0) +O(∆t

2)

I following [Bispen et al., 2017]: (fully-discrete)I L1 estimate for non-linear explicit stepI L2 estimate for linear implicit stepI interpolation between the norms

I follwoing [Gallouët et al., 2017; Feireisl et al., 2016; Berthon et al., 2016; Fischer,2015]?

New applications?I Euler with congestion? (with Charlotte Perrin)

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ConclusionWe have analyzed the RS-IMEX scheme for shallow water equations:

I 1d, 2d, 2d + Coriolis

I “rigorous” asymptotic analysis =⇒ AP!I reasonable numerical results

I H.Z., Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in onespace dimension, HAL: hal-01491450.

I H.Z., Asymptotic consistency of the RS-IMEX scheme for the low-Froude shallow waterequations: Analysis and numerics, XVI International Conference on Hyperbolic Problems.

I H.Z., The RS-IMEX scheme for the rotating shallow water equations with the Coriolis force,In International Conference on Finite Volumes for Complex Applications, pp. 199–207.

Springer, Cham (2017).

Merci de votre attention !

34/34

References

The basic idea: stability of the modified equation

Linear system ∂tU + A∂xU = 0 [Schütz and Noelle, 2014]:

∂tU + A∂xU = Dν∂2xU, Dν :=∆t

2

(α∆x∆t

Iq − Â2 + Ã2 + [Ã, Â])

is stable if P(ξ) := −iAξ − ξ2Dν has only eva with negative real parts.[Ĝ ′, G̃ ′

]=[G ′(U),G ′(U)

]=⇒ smaller, for smaller ‖U −U‖

modified version as in [Zakerzadeh and Noelle, 2016]

P̃(ξ) := −iξΛ− ξ2∆t

2

[α∆x∆t

Iq − Λ2 + 2QR→R̃ Λ̃Q−1R→R̃

Λ].

limε→0‖U −U‖ = 0 =⇒ R and R̃ get closer =⇒ Q

R→R̃ → Iq

1/6

References

T n+11,ij = Dtznij +

1

Θ

(∇h,xm1 ij +∇h,xm2 ij

)n+1T n+12,ij = Dtm1

nij +∇h,x

(m1

2ij

Θz ij − bij

)n+1+∇h,y

(m1 ijm2 ij

Θz ij − bij

)n+1+

1

2 ε∇h,x

(Θz2ij − 2bijz ij

)n+1+

1

εzn+1ij ∇h,xbij −

1

εm2

n+1ij

→ one can check that ‖T n+1∆ ‖ = O(1)

2/6

References

References I

Akio Arakawa. Computational design for long-term numerical integration of the equations of fluid motion:Two-dimensional incompressible flow. Part I. Journal of Computational Physics, 1(1):119–143, 1966.

Emmanuel Audusse, Rupert Klein, and Antony Z. Owinoh. Conservative discretization of Coriolis force in a finitevolume framework. Journal of Computational Physics, 228(8):2934–2950, 2009.

Emmanuel Audusse, Rupert Klein, Duc D. Nguyen, and Stefan Vater. Preservation of the discrete geostrophicequilibrium in shallow water flows, pages 59–67. Springer Berlin Heidelberg, 2011. doi:10.1007/978-3-642-20671-9 7.

Emmanuel Audusse, Stéphane Dellacherie, Pascal Omnes Do Minh Hieu, and Yohan Penel. Godunov type schemefor the linear wave equation with Coriolis source term. HAL: hal-01254888, 2015.

Emmanuel Audusse, Minh Hieu Do, Pascal Omnes, and Yohan Penel. Analysis of modified Godunov type schemesfor the two-dimensional linear wave equation with Coriolis source term on cartesian meshes. HAL: hal-01618753,2017.

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