-
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).
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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
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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
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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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
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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).
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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
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).
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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I homogeneity =⇒ % constantI 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).
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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I homogeneity =⇒ % constantI 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).
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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I homogeneity =⇒ % constantI 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).
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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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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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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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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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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
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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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)
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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)
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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
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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
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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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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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Outline
Introduction
RS-IMEX scheme
1d SWE
2d SWENumerical experiments
2d RSWENumerical experiments
Recent progress
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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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 ‖
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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
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
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Outline
Introduction
RS-IMEX scheme
1d SWE
2d SWENumerical experiments
2d RSWENumerical experiments
Recent progress
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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)
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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
)
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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
(i) solvable: a unique solution for all ε > 0
(ii) “ε-stable”: limε→0‖V n+1∆ ‖ = O(1) (if U = 0 and ∇xη
b = 0)
(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
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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Numerical experiments
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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Numerical experiments
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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Numerical experiments
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
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Numerical experiments
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 .
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Outline
Introduction
RS-IMEX scheme
1d SWE
2d SWENumerical experiments
2d RSWENumerical experiments
Recent progress
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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)
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
∂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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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
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
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
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
(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
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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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
Numerical experiments
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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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
Numerical experiments
Relative error for z.|z∆(t) − z∆(0)| for t = 10 and ε = 10
−4.
=⇒ the scheme is uniformly accurate and AS!
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
Numerical experiments
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!
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
Numerical experiments
Should we invest in U?
ε = 0.1, U = U(0) . ε = 0.1, U = 0 .
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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
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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
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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Introduction RS-IMEX scheme 1d SWE 2d SWE 2d RSWE Recent
progress
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 !
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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
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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)
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References
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the shallow water equations in one spacedimension. HAL:
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Submitted for publication.
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for the low-Froude shallow water equations:Analysis and numerics.
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IntroductionRS-IMEX scheme1d SWE2d SWENumerical experiments
2d RSWENumerical experiments
Recent progress