Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
On all-regime, high-order and well-balanced Lagrange-Projection type schemes for the shallow water equations Maxime Stauffert Université de Versailles Saint-Quentin (UVSQ) and Maison de la Simulation (MdlS) Christophe Chalons, Pierre Kestener, Samuel Kokh and Raphaël Loubère
SHARK-FV, May 2017
1/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Contents
1 2
3
4
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Introduction Continuous equations Shallow Water Equations Operators splitting Relaxation Method FV schemes FV Discretization IMEX properties DG scheme Notations Time discretization Space discretization Transport step
Maxime Stauffert
5
6
7
8
Theoretical results Implicit-explicit DG scheme Mean WB property Nodal WB property Numerical results WB property Dam Break Propagation of perturbation Transcritical regime Limitors MOOD approach Naive approach Robustness approach Conclusion
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Contents
1
2
3
4
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5
Theoretical results
6
Numerical results
7
MOOD approach
8
Conclusion
Introduction Continuous equations FV schemes DG scheme
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
Introduction Construction of a Discontinuous Galerkin (DG) scheme for Shallow Water equations (SWE) Theory based on Finite Volume (FV) Lagrange-Projection (L-P) type schemes for Euler equations1 and for SWE2 Low Froude number : fast acoustic waves vs. slow material transport waves Acoustic - Transport operators decomposition (L-P like) : −→ Impliciting fast phenomenons : less restrictive CFL condition −→ Expliciting slow phenomenons : reasonable precision
1 Christophe Chalons, Mathieu Girardin, and Samuel Kokh. “An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes”. In: Communications in Computational Physics 20.01 (2016), pp. 188–233. 2 Christophe Chalons et al. “A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations”. In: Communic. Math. Sci. 15.3 (2017), pp. 765–788. 4/33
Maxime Stauffert
LP type schemes for SWE
Introduction
Continuous equations
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DG scheme
Theoretical results
Numerical results
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Contents
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2
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Theoretical results
6
Numerical results
7
MOOD approach
8
Conclusion
Introduction Continuous equations Shallow Water Equations Operators splitting Relaxation Method
3
FV schemes
4
DG scheme
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Shallow Water Equations Euler System in 1D
∂t ρ + ∂x (ρu) = 0, ∂t (ρu) + ∂x (ρu 2 + p) = 0, ∂t (ρE ) + ∂x ((ρE + p)u) = 0.
Shallow Water Sytem in 1D
∂t h + ∂x (hu) = 0, h2 2 = −gh∂x z. ∂t (hu) + ∂x hu + g 2
−→ Two similar systems −→ Non-conservative source term in SWE
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Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Operators splitting
"Acoustic" / "Transport" decomposition
∂t h+
∂t (hu)+
7/33
hu ∂x u + ∂x
Maxime Stauffert
h ∂x u+ u ∂x h = 0, 2 h + u ∂x (hu) = −gh∂x z. g 2
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Operators splitting
"Acoustic" / "Transport" decomposition
Acoustic − t n →t n+1
Transport t n+1− →t n+1
7/33
∂t h+
∂t (hu)+ ( ∂t h+ ∂t (hu)+
h ∂x u = 0, 2 h = −gh∂x z, hu ∂x u + ∂x g 2 u ∂x h = 0, u ∂x (hu) = 0.
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Relaxation Method Change of variable : h −→ τ = 1/h Approximation of τ (·, t) ∂x X by τ (·, t n )∂x X = ∂m X Variable π : linearisation of the pressure
g 2τ 2
Acoustic System
∂t h+
∂t (hu)+
8/33
hu ∂x u + ∂x
Maxime Stauffert
h ∂x u = 0, h2 g = −gh∂x z, 2
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Relaxation Method Change of variable : h −→ τ = 1/h Approximation of τ (·, t) ∂x X by τ (·, t n )∂x X = ∂m X Variable π : linearisation of the pressure
g 2τ 2
Acoustic System
∂t τ − τ ∂x u = 0, g = −g∂x z, ∂t u + τ ∂x 2τ 2 ∂t z = 0.
8/33
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Relaxation Method Change of variable : h −→ τ = 1/h Approximation of τ (·, t) ∂x X by τ (·, t n )∂x X = ∂m X Variable π : linearisation of the pressure
g 2τ 2
Acoustic System
∂t τ − ∂m u = 0, g g = − ∂m z, ∂t u + ∂m 2 2τ τ ∂t z = 0.
8/33
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Relaxation Method Change of variable : h −→ τ = 1/h Approximation of τ (·, t) ∂x X by τ (·, t n )∂x X = ∂m X Variable π : linearisation of the pressure
g 2τ 2
Relaxed Acoustic System
∂t τ − ∂m u = 0,
g ∂t u + ∂m π = − ∂m z, τ π − 2τg 2 2 ∂ π + a ∂ u = − , t m ε ∂t z = 0.
8/33
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Relaxation Method Change of variable : h −→ τ = 1/h Approximation of τ (·, t) ∂x X by τ (·, t n )∂x X = ∂m X Variable π : linearisation of the pressure
g 2τ 2
Relaxed Acoustic System
∂t τ − ∂m u = 0,
g ∂t u + ∂m π = − ∂m z, τ π − 2τg 2 2 ∂ π + a ∂ u = − , t m ε ∂t z = 0. Prop : Viscous approximation of the Acoustic systempunder the sub-characteristic condition : a > max(hc) = max( τ1 τg ). 8/33
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Relaxation Method Operators splitting : Instantaneous relaxation step Homogeneous relaxed Acoustic system Relaxed Acoustic System ∂t τ = 0, ∂t u = 0,
g
π − 2τ 2 , ∂t π = − ε ∂t z = 0,
8/33
and
Maxime Stauffert
∂t τ − ∂m u ∂t u + ∂m π + g ∂m z τ 2 ∂ π + a ∂m u t ∂t z
LP type schemes for SWE
= 0, = 0, = 0, = 0.
References
Introduction
Continuous equations
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DG scheme
Theoretical results
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Contents
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Theoretical results
6
Numerical results
7
MOOD approach
8
Conclusion
Introduction Continuous equations FV schemes FV Discretization IMEX properties DG scheme
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
FV Discretization Acoustic step − ∆t n ∗,α ∗,α n τj uj+1/2 − uj−1/2 = Lα τjn+1 = τjn + j τj , ∆x − ∆t n ∗,α n ∗,α ujn+1 = ujn − τj πj+1/2 − πj−1/2 − ∆t τjn {gh∂x z}j , ∆x π n+1− = π n − a2 ∆t τ n u ∗,α − u ∗,α j j j j j+1/2 j−1/2 . ∆x Transport step ∆t ∗,n+1− ∗,α ∗,n+1− ∗,α n+1− hjn+1 = Lα − hj+1/2 uj+1/2 − hj−1/2 uj−1/2 , j hj ∆x − (hu)n+1 = Lα (hu)n+1 − ∆t (hu)∗,n+1− u ∗,α − (hu)∗,n+1− u ∗,α j j j j+1/2 j+1/2 j−1/2 j−1/2 . ∆x α = n (full explicit scheme) or n + 1− (implicit-explicit scheme) 10/33
Maxime Stauffert
LP type schemes for SWE
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
FV Discretization Acoustic step − ∆t n ∗,α ∗,α n τj uj+1/2 − uj−1/2 = Lα τjn+1 = τjn + j τj , ∆x − ∆t n ∗,α n ∗,α ujn+1 = ujn − τj πj+1/2 − πj−1/2 − ∆t τjn {gh∂x z}j , ∆x π n+1− = π n − a2 ∆t τ n u ∗,α − u ∗,α j j j j j+1/2 j−1/2 . ∆x Transport step ∆t ∗,n+1− ∗,α ∗,n+1− ∗,α n+1− hjn+1 = Lα − hj+1/2 uj+1/2 − hj−1/2 uj−1/2 , j hj ∆x − (hu)n+1 = Lα (hu)n+1 − ∆t (hu)∗,n+1− u ∗,α − (hu)∗,n+1− u ∗,α j j j j+1/2 j+1/2 j−1/2 j−1/2 . ∆x α = n (full explicit scheme) or n + 1− (implicit-explicit scheme) 10/33
Maxime Stauffert
LP type schemes for SWE
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
IMEX properties Hypothesis : Subcharacteristic condition : a > maxj (hj cj ) ∗ 1 ∆t CFL condition : ∆x maxj uj+1/2 ≤ 2 Properties : Conservative for h (and for hu if z = cst) hjn > 0, ∀j, n, provided that hj0 > 0, ∀j. Degeneration to classical L-P scheme if z = cst ({gh∂x z} = 0) Well-balanced : preservation of the "lake at rest" conditions (u = 0 and h + z = cst) It satisfies a discrete entropy inequality of the form : Ujn+1 − Ujn + 11/33
∆t n+1− n+1− Fj+1/2 − Fj−1/2 ≤ −∆t {ghu∂x z}j ∆xj Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
IMEX properties Hypothesis : Subcharacteristic condition : a > maxj (hj cj ) ∗ 1 ∆t CFL condition : ∆x maxj uj+1/2 ≤ 2 Properties : Conservative for h (and for hu if z = cst) hjn > 0, ∀j, n, provided that hj0 > 0, ∀j. Degeneration to classical L-P scheme if z = cst ({gh∂x z} = 0) Well-balanced : preservation of the "lake at rest" conditions (u = 0 and h + z = cst) It satisfies a discrete entropy inequality of the form : Ujn+1 − Ujn + 11/33
∆t n+1− n+1− Fj+1/2 − Fj−1/2 ≤ −∆t {ghu∂x z}j ∆xj Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Contents
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Theoretical results
6
Numerical results
7
MOOD approach
8
Conclusion
Introduction Continuous equations FV schemes DG scheme Notations Time discretization Space discretization Transport step
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
Notations Based on work from Florent Renac3 at ONERA Wrote for SWE without topography4 Lagrange polynomials on Gauss-Lobatto quadrature: ρ(x ) =
p X
ρk,j φk,j (x ),
∀x ∈ xj−1/2 , xj+1/2
k=0
2 with φk,j (x ) = `k ∆x (x − xj ) , `k (si ) = δk,i and si are the Gauss-Lobatto quadrature points on [−1, 1] Numerical integration on the same Gauss-Lobatto quadrature points: xj+1/2
Z f (x ) dx ' xj−1/2
p p ∆x X ∆x ∆x X ωk f (xk,j ) = ωk f xj + sk 2 2 2 k=0
k=0
3 Florent Renac. “A robust high-order Lagrange-projection like scheme with large time steps for the isentropic Euler equations”. In: Numerische Mathematik (2016), pp. 1–27. 4 Christophe Chalons and Maxime Stauffert. “A high-order Discontinuous Galerkin Lagrange-Projection scheme for the barotropic Euler equations”. In: To appear in FVCA8 conference proceedings (2017). 13/33
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LP type schemes for SWE
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Time discretization Multiplication by a Lagrange polynomial and integration over a cell Development of the derivatives in time Numerical integration of the Lagrange polynomials Discretization of the derivatives in time Homogeneous relaxed Acoustic system
14/33
∂t τ − ∂m u = 0, g ∂t u + ∂m π = − ∂m z, τ 2 ∂t π + a ∂m u = 0.
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Time discretization Multiplication by a Lagrange polynomial and integration over a cell Development of the derivatives in time Numerical integration of the Lagrange polynomials Discretization of the derivatives in time Homogeneous relaxed Acoustic system
Z
Z φi,j ∂t τ dx −
φi,j ∂m u dx = 0,
κj
κj
Z
Z
Z φi,j ∂m π dx = −
φi,j ∂t u dx +
κj κ Z Z j φi,j ∂t π dx + a2 φi,j ∂m u dx = 0. κj
14/33
φi,j κj
κj
Maxime Stauffert
LP type schemes for SWE
g ∂m z, τ
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Time discretization Multiplication by a Lagrange polynomial and integration over a cell Development of the derivatives in time Numerical integration of the Lagrange polynomials Discretization of the derivatives in time Homogeneous relaxed Acoustic system
Z p X k=0 p X
! φi,j φk,j dx
Z ∂t τk,j −
φi,j ∂m u dx = 0,
κj
κj
!
Z φi,j φk,j dx
Z
Z φi,j ∂m π dx = −
∂t uk,j +
κj κj k=0 ! Z Z p X φi,j φk,j dx ∂t πk,j + a2 φi,j ∂m u dx = 0. k=0
14/33
κj
κj
Maxime Stauffert
LP type schemes for SWE
φi,j κj
g ∂m z, τ
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Time discretization Multiplication by a Lagrange polynomial and integration over a cell Development of the derivatives in time Numerical integration of the Lagrange polynomials Discretization of the derivatives in time Homogeneous relaxed Acoustic system
∆x ωi ∂t τi,j − 2
Z
∆x ωi ∂t ui,j + 2
Z
φi,j ∂m u dx = 0, κj
Z φi,j ∂m π dx = −
κj Z ∆x ωi ∂t πi,j + a2 φi,j ∂m u dx = 0. 2 κj
14/33
Maxime Stauffert
φi,j κj
LP type schemes for SWE
g ∂m z, τ
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Time discretization Multiplication by a Lagrange polynomial and integration over a cell Development of the derivatives in time Numerical integration of the Lagrange polynomials Discretization of the derivatives in time Homogeneous relaxed Acoustic system
Z 2∆t n+1− n = τ φi,j ∂m u α dx , τ + i,j i,j ω ∆x i κj ! Z Z 2∆t g n+1− n α ui,j = ui,j − φi,j ∂m π dx + φi,j n ∂m z dx , ωi ∆x τ κ κ j j Z 2∆t n+1− n = πi,j − a2 φi,j ∂m u α dx . πi,j ωi ∆x κj
14/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Space discretization How to write equation on τ as in finite volume ? Approximation of the integral of ∂m u Integration by part (exact) Introduction of the numerical fluxes Treatment of the source term Homogeneous relaxed Acoustic system
=
n τi,j
2∆t + ωi ∆x
Z
φi,j ∂m u α dx
κj
←→
n+1− τi,j
−
τjn+1 = τjn +
15/33
∆t n ∗,α ∗,α n τj uj+1/2 − uj−1/2 = Lα j τj ∆x
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Space discretization How to write equation on τ as in finite volume ? Approximation of the integral of ∂m u Integration by part (exact) Introduction of the numerical fluxes Treatment of the source term Homogeneous relaxed Acoustic system
Z κj
15/33
φi,j ∂m u α dx '
∆x n α n ωi τi,j ∂x ui,j = τi,j 2
Maxime Stauffert
Z
φi,j ∂x u α dx
κj
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Space discretization How to write equation on τ as in finite volume ? Approximation of the integral of ∂m u Integration by part (exact) Introduction of the numerical fluxes Treatment of the source term Homogeneous relaxed Acoustic system
Z κj
Z ∆x n α n ωi τi,j ∂x ui,j = τi,j φi,j ∂x u α dx 2 κj ! Z n α α ' τi,j [φi,j u ] − u ∂x φi,j dx
φi,j ∂m u α dx '
κj
15/33
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Space discretization How to write equation on τ as in finite volume ? Approximation of the integral of ∂m u Integration by part (exact) Introduction of the numerical fluxes Treatment of the source term Homogeneous relaxed Acoustic system
Z κj
Z ∆x n α n ωi τi,j ∂x ui,j = τi,j φi,j ∂x u α dx 2 κj ! Z n α α ' τi,j [φi,j u ] − u ∂x φi,j dx
φi,j ∂m u α dx '
κj n ' τi,j
∗,α δi,p uj+1/2
−
∗,α δi,0 uj−1/2
−
p X k=0
15/33
Maxime Stauffert
LP type schemes for SWE
! α ωk uk,j ∂x `i (sk )
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Space discretization How to write equation on τ as in finite volume ? Approximation of the integral of ∂m u Integration by part (exact) Introduction of the numerical fluxes Treatment of the source term Homogeneous relaxed Acoustic system
Z φi,j κj
g ∆x n g n ∂m z ' ωi τi,j n ∂x z ' τi,j n τ 2 τi,j n −→ τi,j δi,p
15/33
Z
φi,j ghn ∂x z
κj
∆x ∆x n n {gh∂x z}j+1/2 + δi,0 {gh∂x z}j−1/2 2 2 ! ∆x n + ωi ghi,j ∂x z|i,j 2
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
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DG scheme
Theoretical results
Numerical results
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Conclusion
Space discretization How to write equation on τ as in finite volume ? Approximation of the integral of ∂m u Integration by part (exact) Introduction of the numerical fluxes Treatment of the source term Homogeneous relaxed Acoustic system
Z 2∆t n n+1− n n τi,j = τi,j + τ φi,j ∂x u α dx = Lα i,j τi,j , ωi ∆x i,j κj ! Z Z 2∆t n n+1− n α n = ui,j − ui,j τ φi,j ∂x π dx + φi,j gh ∂x z dx , ωi ∆x i,j κj κj Z 2∆t n n+1− n τi,j φi,j ∂x u α dx . = πi,j − a2 πi,j ωi ∆x κj 15/33
Maxime Stauffert
LP type schemes for SWE
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Introduction
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Conclusion
Transport step How to write equation on X = h, hu as in finite volume ? Rewriting the integral of u ∂x X Approximation of the integral of X ∂x u α to bring out Lα i,j Integration by part (not exact) Introduction of the numerical fluxes Transport system =
n+1− Xi,j
2∆t − ωi ∆x
Z
−
u α φi,j ∂x X n+1 dx
κj
←→
n+1 Xi,j
∆t ∗,n+1− ∗,α ∗,n+1− ∗,α uj−1/2 Xj+1/2 uj+1/2 − Xj+1/2 ∆x ( − ∗,α Xjn+1 , if uj+1/2 ≥ 0, ∗,n+1− with Xj+1/2 = − n+1 Xj+1 , otherwise. −
n+1 Xjn+1 = Lα − j Xj
16/33
Maxime Stauffert
LP type schemes for SWE
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Introduction
Continuous equations
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Theoretical results
Numerical results
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Conclusion
Transport step How to write equation on X = h, hu as in finite volume ? Rewriting the integral of u ∂x X Approximation of the integral of X ∂x u α to bring out Lα i,j Integration by part (not exact) Introduction of the numerical fluxes Transport system Z
α
u φi,j ∂x X κj
16/33
n+1−
Z =
φi,j ∂x (X κj
Maxime Stauffert
n+1− α
Z
u )− κj
LP type schemes for SWE
−
X n+1 φi,j ∂x u α
References
Introduction
Continuous equations
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DG scheme
Theoretical results
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Conclusion
Transport step How to write equation on X = h, hu as in finite volume ? Rewriting the integral of u ∂x X Approximation of the integral of X ∂x u α to bring out Lα i,j Integration by part (not exact) Introduction of the numerical fluxes Transport system Z
α
u φi,j ∂x X
n+1−
Z =
κj
φi,j ∂x (X
n+1− α
u )−
κj
Z '
Z κj
−
−
n+1 φi,j ∂x (X n+1 u α ) − Xi,j
κj
16/33
−
X n+1 φi,j ∂x u α
Maxime Stauffert
Z κj
LP type schemes for SWE
φi,j ∂x u α
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Transport step How to write equation on X = h, hu as in finite volume ? Rewriting the integral of u ∂x X Approximation of the integral of X ∂x u α to bring out Lα i,j Integration by part (not exact) Introduction of the numerical fluxes Transport system Z
α
u φi,j ∂x X
n+1−
Z =
κj
φi,j ∂x (X
n+1− α
u )−
κj
Z '
Z
−
X n+1 φi,j ∂x u α
κj −
−
n+1 φi,j ∂x (X n+1 u α ) − Xi,j
Z
κj
φi,j ∂x u α
κj
Z h i Z − − n+1− ' φi,j X n+1 u α − X n+1 u α ∂x φi,j − Xi,j φi,j ∂x u α κj
16/33
Maxime Stauffert
κj
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Transport step How to write equation on X = h, hu as in finite volume ? Rewriting the integral of u ∂x X Approximation of the integral of X ∂x u α to bring out Lα i,j Integration by part (not exact) Introduction of the numerical fluxes Transport system Z
α
u φi,j ∂x X
n+1−
Z =
κj
φi,j ∂x (X
n+1− α
u )−
κj
Z '
Z
−
X n+1 φi,j ∂x u α
κj −
−
n+1 φi,j ∂x (X n+1 u α ) − Xi,j
Z
κj
φi,j ∂x u α
κj
Z h i Z − − n+1− ' φi,j X n+1 u α − X n+1 u α ∂x φi,j − Xi,j φi,j ∂x u α κj
h
−
κj −
−
i
∗,n+1 ∗,α ∗,n+1 ∗,α −→ φi,j X n+1 u α = δi,p Xj+1/2 uj+1/2 − δi,0 Xj+1/2 uj−1/2
16/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Transport step How to write equation on X = h, hu as in finite volume ? Rewriting the integral of u ∂x X Approximation of the integral of X ∂x u α to bring out Lα i,j Integration by part (not exact) Introduction of the numerical fluxes Transport system
n+1− n+1 hi,j = Ln+1− hi,j − i,j
2∆t wi ∆x
Z κj
2∆t n+1− n+1− n+1 − (hu)i,j = Li,j (hu)i,j wi ∆x
16/33
Maxime Stauffert
−
φi,j ∂x (hn+1 u α ) dx , Z
−
φi,j ∂x ((hu)n+1 u α ) dx .
κj
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Transport step How to write equation on X = h, hu as in finite volume ? Rewriting the integral of u ∂x X Approximation of the integral of X ∂x u α to bring out Lα i,j Integration by part (not exact) Introduction of the numerical fluxes Transport system
16/33
n+1 n hi,j = hi,j −
2∆t wi ∆x
n (hu)n+1 i,j = (hu)i,j −
Z
−
φi,j ∂x hn+1 u α dx ,
κj
2∆t wi ∆x
Z
−
φi,j ∂x ((hu)n+1 u α + π α ) dx
κj
Z +
! φi,j ghn ∂x z dx .
κj
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Contents
17/33
1
Introduction
2
Continuous equations
3
FV schemes
4
DG scheme
Maxime Stauffert
5
Theoretical results Implicit-explicit DG scheme Mean WB property Nodal WB property
6
Numerical results
7
MOOD approach
8
Conclusion
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Implicit-explicit DG scheme Hypothesis : a > maxj maxi hi,j
p
ghi,j
∆t ∆x
maxj maxi ci,j ≤ 1 R ∗ ∗ + δi,0 uj−1/2,+ with ci,j = ω2i κj ujn+1− ∂x φi,j − δi,p uj+1/2,− Properties ∗ ∗ If p = 0 : cj = uj−1/2,+ − uj+1/2,− −→ same CFL as in FV
Convex combination : n+1
Xj
=
Pp
ωi i=0 2
1− +
−
n+1
n+1 hi,j > 0 and thus hj
18/33
n+1− ∆t ∆x ci,j Xi,j n+1− ∆t ∗ ∆x (−uj+1/2,− )X0,j+1
+
n+1− ∆t ∗ ∆x uj−1/2,+ Xp,j−1
n > 0, provided that hi,j > 0, ∀i, j
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Implicit-explicit DG scheme Hypothesis : a > maxj maxi hi,j
p
ghi,j
∆t ∆x
maxj maxi ci,j ≤ 1 R ∗ ∗ + δi,0 uj−1/2,+ with ci,j = ω2i κj ujn+1− ∂x φi,j − δi,p uj+1/2,− Properties ∗ ∗ If p = 0 : cj = uj−1/2,+ − uj+1/2,− −→ same CFL as in FV
Convex combination : n+1
Xj
=
Pp
ωi i=0 2
1− +
−
n+1
n+1 hi,j > 0 and thus hj
18/33
n+1− ∆t ∆x ci,j Xi,j n+1− ∆t ∗ ∆x (−uj+1/2,− )X0,j+1
+
n+1− ∆t ∗ ∆x uj−1/2,+ Xp,j−1
n > 0, provided that hi,j > 0, ∀i, j
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Implicit-explicit DG scheme Hypothesis : a > maxj maxi hi,j
p
ghi,j
∆t ∆x
maxj maxi ci,j ≤ 1 R ∗ ∗ + δi,0 uj−1/2,+ with ci,j = ω2i κj ujn+1− ∂x φi,j − δi,p uj+1/2,− Properties It satisfies a discrete entropy inequality of the form : n+1
(hE )(U j
n
) − (hE )j +
∆t h ∗ ∗ ((hE )∗j+1/2 + πj+1/2 )uj+1/2 ∆x ∗ ∗ − ((hE )∗j−1/2 + πj−1/2 )uj−1/2
i
≤ −∆t {ghu∂x z}j . 18/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Mean WB property Hypothesis : h0 + z 0 = K and u 0 = 0 with h0 and z 0 polynomials of order ≤ p
Properties
19/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Mean WB property Hypothesis : h0 + z 0 = K and u 0 = 0 with h0 and z 0 polynomials of order ≤ p
Properties n+1
hu j
n
∗,n ∗,n ∆t 2 hu + π j+1/2 − hu 2 + π j−1/2 ∆x ! Z ∆t − ∆x {gh∂x z}j + gh ∂x z ∆x κj
= hu j −
∗ ∗ with hu 2 = 0, πj+1/2 − πj−1/2 + ∆x {gh∂x z}j = πp,j − π0,j and
Z
Z gh ∂x z = −
κj
19/33
g∂x κj
Maxime Stauffert
h2 = − [π] 2
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Mean WB property Hypothesis : h0 + z 0 = K and u 0 = 0 with h0 and z 0 polynomials of order ≤ p
Properties n+1
hu j
n
= hu j
−→ WB for the mean values and only for the EXEX scheme 19/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Nodal WB property Hypothesis : h0 + z 0 = K and u 0 = 0 with h0 and z 0 polynomials of order ≤ p/2
Properties h2 = − [π] + 2 κj κj 0 τ The only solution of the linear system is u 0 π0 Z
φi,j ghn ∂x z = −
Z
Z
φi,j g∂x
π n ∂x φi,j
κj
−→ WB for the nodal values the EXEX and the IMEX schemes
20/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Nodal WB property Hypothesis : h0 + z 0 = K and u 0 = 0 with h0 and z 0 polynomials of order ≤ p/2
Properties h2 = − [π] + 2 κj κj 0 τ The only solution of the linear system is u 0 π0 Z
φi,j ghn ∂x z = −
Z
Z
φi,j g∂x
π n ∂x φi,j
κj
−→ WB for the nodal values the EXEX and the IMEX schemes
20/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Contents
1
21/33
5
Theoretical results
6
Numerical results WB property Dam Break Propagation of perturbation Transcritical regime Limitors
7
MOOD approach
8
Conclusion
Introduction
2
Continuous equations
3
FV schemes
4
DG scheme
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
WB property 25
Total water height H = h+z Topography : z
Heights : H, z
20 15 10 5 0
0
200
400
600
800
1000
x
22/33
Maxime Stauffert
LP type schemes for SWE
1200
1400
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
WB property
h and z of order 2 N = 500
EXEX
IMEX
22/33
n=1 kh+z−15k∞/15
T = 20 kq/hk∞
kh+z−15k∞/15
kq/hk∞
p=0
2.37 E−16
0.00 E−16
2.37 E−16
0.00 E−16
p=1
1.18 E−16
3.19 E−16
4.72 E−2
1.46 E+0
p=2
2.37 E−16
1.89 E−16
6.62 E−3
1.92 E−1
p=3
2.37 E−16
1.78 E−16
3.76 E−4
6.01 E−3
p=4
2.37 E−16
0.00 E−16
2.37 E−16
0.00 E−16
p=0
2.37 E−16
0.00 E−16
2.37 E−16
0.00 E−16
p=1
4.68 E−1
3.04 E+1
9.09 E−1
9.31 E+1
p=2
1.79 E−2
4.53 E−1
4.94 E−2
4.64 E−1
p=3
1.33 E−3
4.68 E−2
3.97 E−3
4.79 E−2
p=4
2.37 E−16
0.00 E−16
2.37 E−16
0.00 E−16
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Dam Break NbCell : 1500, Tf : 50 25
Heights : H, z
20 15 ACU EXEX0 EXEX1 EXEX2 Topography : z
10 5 0
0
200
400
600
800
1000
1200
x 23/33
Maxime Stauffert
LP type schemes for SWE
1400
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Dam Break NbCell : 1500, Tf : 50 17.5
ACU EXEX0 EXEX1 EXEX2
Total height : H
17 16.5 16 15.5 15 14.5 1340
1350
1360
1370
1380
x 23/33
Maxime Stauffert
LP type schemes for SWE
1390
1400
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Dam Break NbCell : 1500, Tf : 50 17.5
ACU IMEX0 IMEX1 IMEX2
Total height : H
17 16.5 16 15.5 15 14.5 1340
1350
1360
1370
1380
x 23/33
Maxime Stauffert
LP type schemes for SWE
1390
1400
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Propagation of perturbation NbCell : 1000, Tf : 0.2 0.002 0.0015 Velocity : u
0.001 0.0005 0 ACU EXEX0 EXEX1 EXEX2
-0.0005 -0.001 -0.0015 -0.002
0
0.5
1
1.5
x 24/33
Maxime Stauffert
LP type schemes for SWE
2
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Propagation of perturbation NbCell : 1000, Tf : 0.2 0.002 0.0015 Velocity : u
0.001 0.0005 0 ACU IMEX0 IMEX1 IMEX2
-0.0005 -0.001 -0.0015 -0.002
0
0.5
1
1.5
x 24/33
Maxime Stauffert
LP type schemes for SWE
2
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Transcritical regime NbCell : 1600, Tf : 200 3.25 3.2 3.15 3.1
HLLACU z EXEXloc IMEXloc
h+z
3.05 3
ref HRHLL
2.95 2.9 2.85 2.8 2.75
25/33
0
5
10
Maxime Stauffert
15
LP type schemes for SWE
20
25
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Limitors NbCell = 500, Tf = 20, p = 2 25
Conserved Characteristic MOOD
height h
20 15 10 5 0
0
200
400
600
800
1000
x 26/33
Maxime Stauffert
LP type schemes for SWE
1200
1400
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Contents
1
2
27/33
5
Theoretical results
6
Numerical results
7
MOOD approach Naive approach Robustness approach
8
Conclusion
Introduction Continuous equations
3
FV schemes
4
DG scheme
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
Naive approach
DG at t n
- 1 DG step
DG at t n+1 ? Projection FV at t n+1 ? Detectors
bad @ good @ - Reconstruct. 1 FV step FV at t n FV at t n+1
28/33
Maxime Stauffert
LP type schemes for SWE
DG at t n+1
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
Naive approach
? - 1 DG step DG at t n
DG at t n+1 ? Projection FV at t n+1 ? Detectors
bad @ good @ - Reconstruct. 1 FV step FV at t n FV at t n+1
28/33
Maxime Stauffert
LP type schemes for SWE
DG at t n+1
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
Naive approach
- 1 DG step
DG at t n @ @ R @
DG at t n+1 ? Projection
Projection @ @@ R
FV at t n+1 ? Detectors
bad @ good @ - Reconstruct. 1 FV step FV at t n FV at t n+1 Problem : Reconstruction ◦ Projection 6= Identity 28/33
Maxime Stauffert
LP type schemes for SWE
DG at t n+1
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Robustness approach
FV at t n
- Reconstruct.
DG at t n - DG steps
DG at t n+1 ? Projection FV at t n+1 ? Detectors bad
FV steps
@ good @ FV at t n+1
29/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Robustness approach
? - Reconstruct. FV at t n
DG at t n - DG steps
DG at t n+1 ? Projection FV at t n+1 ? Detectors bad
FV steps
@ good @ FV at t n+1
29/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Robustness approach
- Reconstruct.
FV at t n PP PP
PP PP P
DG at t n - DG steps
DG at t n+1 ? Projection
PP P
PP P
PP P
FV at t n+1 ? Detectors
bad FV steps ?
@ good @ FV at t n+1
29/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Robustness approach
FV at t n
? DG at t n - Reconstruct. - DG steps
DG at t n+1 ? Projection FV at t n+1 ? Detectors bad
FV steps
Projection ◦ Reconstruction = Identity
@ good @ FV at t n+1
⇒ no recomputation of DG solution when detector = 0 29/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Contents
1
2
3
4
30/33
5
Theoretical results
6
Numerical results
7
MOOD approach
8
Conclusion
Introduction Continuous equations FV schemes DG scheme
Maxime Stauffert
LP type schemes for SWE
Conclusion
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
Conclusion Achievements DG discretization for L-P schemes in framework of SWE Well-balanced properties More robust results with MOOD Implementation of a compiled code Perspectives Rework of the code for the Robust MOOD approach Multi-dimensional system Study of low Froude flows for those schemes Study other systems that have some asymptotic regime (eg. MHD) Use those schemes with AMR techniques in CanoP
31/33
Maxime Stauffert
LP type schemes for SWE
References
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
MOOD approach
Conclusion
References
Bibliography Christophe Chalons, Mathieu Girardin, and Samuel Kokh. “An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes”. In: Communications in Computational Physics 20.01 (2016), pp. 188–233. Christophe Chalons and Maxime Stauffert. “A high-order Discontinuous Galerkin Lagrange-Projection scheme for the barotropic Euler equations”. In: To appear in FVCA8 conference proceedings (2017). Christophe Chalons et al. “A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations”. In: Communic. Math. Sci. 15.3 (2017), pp. 765–788. Florent Renac. “A robust high-order Lagrange-projection like scheme with large time steps for the isentropic Euler equations”. In: Numerische Mathematik (2016), pp. 1–27.
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Maxime Stauffert
LP type schemes for SWE
Introduction
Continuous equations
FV schemes
DG scheme
Theoretical results
Numerical results
Thank you for your attention
33/33
Maxime Stauffert
LP type schemes for SWE
MOOD approach
Conclusion
References
