A high-order, admissibility and asymptotic-preserving ...€¦ · Outline 1...
Transcript of A high-order, admissibility and asymptotic-preserving ...€¦ · Outline 1...
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A high-order, admissibility and asymptotic-preservingfinite volume scheme on 2D unstructured meshes
Laboratoire deMathématiquesJeanLeray
UMR 6629 - Nantes
F. Blachère1, R. Turpault2
1Laboratoire de Mathématiques Jean Leray (LMJL),Université de Nantes,
2Institut de Mathématiques de Bordeaux (IMB),Bordeaux-INP
SHARK-FV,24/05/2016, Póvoa de Varzim (Portugal)
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV scheme
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 1/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV scheme
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Problematic
Hyperbolic systems of conservation laws with source term:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
A: set of admissible states,W ∈ A ⊂ RN ,F: physical flux,γ > 0: controls the stiffness,R: A → A ; smooth function with some compatibility conditions(Berthon, LeFloch, and Turpault [BLT13]).
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 2/26
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Problematic
Hyperbolic systems of conservation laws with source term:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
A: set of admissible states,W ∈ A ⊂ RN ,F: physical flux,γ > 0: controls the stiffness,R: A → A ; smooth function with some compatibility conditions(Berthon, LeFloch, and Turpault [BLT13]).
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 2/26
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Problematic
Hyperbolic systems of conservation laws with source term:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
Under compatibility conditions on R, when γt→∞, (1) degenerates into adiffusion equation:
∂tw − div(f(w)∇w
)= 0 (2)
w ∈ R, linked to W,f(w) > 0.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 3/26
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Example: isentropic Euler equations with friction
{∂tρ+ div(ρu) = 0
∂tρu + div(ρu⊗ u) +∇p = −κρu , with: p′(ρ) > 0
A = {(ρ, ρu)T ∈ R3/ρ > 0}
Formalism of (1):
W = (ρ ρu)T
R(W) = (ρ 0)T
F(W) =(ρu ρu⊗ u + pI
)Tγ(W) = κ > 0
Limit diffusion equation:
∂tρ− div(p′(ρ)
κ∇ρ)
= 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 4/26
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Example: isentropic Euler equations with friction
{∂tρ+ div(ρu) = 0
∂tρu + div(ρu⊗ u) +∇p = −κρu , with: p′(ρ) > 0
A = {(ρ, ρu)T ∈ R3/ρ > 0}
Formalism of (1):
W = (ρ ρu)T
R(W) = (ρ 0)T
F(W) =(ρu ρu⊗ u + pI
)Tγ(W) = κ > 0
Limit diffusion equation:
∂tρ− div(p′(ρ)
κ∇ρ)
= 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 4/26
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Example: isentropic Euler equations with friction
{∂tρ+ div(ρu) = 0
∂tρu + div(ρu⊗ u) +∇p = −κρu , with: p′(ρ) > 0
A = {(ρ, ρu)T ∈ R3/ρ > 0}
Formalism of (1):
W = (ρ ρu)T
R(W) = (ρ 0)T
F(W) =(ρu ρu⊗ u + pI
)Tγ(W) = κ > 0
Limit diffusion equation:
∂tρ− div(p′(ρ)
κ∇ρ)
= 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 4/26
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Goal of an AP scheme
Conservation laws (1):∂tW + div(F(W)) = γ(W)(R(W)−W)
Diffusion equation (2):∂tw − div
(f(w)∇w
)= 0
γt→∞
Numerical scheme
consistent:∆t,∆x→ 0
Limit schemeγt→∞
consistent?
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 5/26
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Goal of an AP scheme
Conservation laws (1):∂tW + div(F(W)) = γ(W)(R(W)−W)
Diffusion equation (2):∂tw − div
(f(w)∇w
)= 0
γt→∞
Numerical scheme
consistent:∆t,∆x→ 0
Limit schemeγt→∞
consistent?
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 5/26
![Page 12: A high-order, admissibility and asymptotic-preserving ...€¦ · Outline 1 Generalcontextandexample 2 Developmentofanadmissibility&asymptoticpreservingFVscheme 3 High-orderextension](https://reader036.fdocuments.us/reader036/viewer/2022071419/6117187a20520726a955fe04/html5/thumbnails/12.jpg)
Goal of an AP scheme
Conservation laws (1):∂tW + div(F(W)) = γ(W)(R(W)−W)
Diffusion equation (2):∂tw − div
(f(w)∇w
)= 0
γt→∞
Numerical scheme
consistent:∆t,∆x→ 0
Limit schemeγt→∞
consistent?
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 5/26
![Page 13: A high-order, admissibility and asymptotic-preserving ...€¦ · Outline 1 Generalcontextandexample 2 Developmentofanadmissibility&asymptoticpreservingFVscheme 3 High-orderextension](https://reader036.fdocuments.us/reader036/viewer/2022071419/6117187a20520726a955fe04/html5/thumbnails/13.jpg)
Goal of an AP scheme
Conservation laws (1):∂tW + div(F(W)) = γ(W)(R(W)−W)
Diffusion equation (2):∂tw − div
(f(w)∇w
)= 0
γt→∞
Numerical scheme
consistent:∆t,∆x→ 0
Limit schemeγt→∞
consistent?
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 5/26
![Page 14: A high-order, admissibility and asymptotic-preserving ...€¦ · Outline 1 Generalcontextandexample 2 Developmentofanadmissibility&asymptoticpreservingFVscheme 3 High-orderextension](https://reader036.fdocuments.us/reader036/viewer/2022071419/6117187a20520726a955fe04/html5/thumbnails/14.jpg)
Goal of an AP scheme
Conservation laws (1):∂tW + div(F(W)) = γ(W)(R(W)−W)
Diffusion equation (2):∂tw − div
(f(w)∇w
)= 0
γt→∞
Numerical scheme
consistent:∆t,∆x→ 0
Limit schemeγt→∞
consistent?
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 5/26
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Example of a non AP scheme in 1D
xi−1 xi xi+1
xi−1/2 xi+1/2xi−3/2 xi+3/2
xi−2 xi+2
∆x
Naïve scheme in 1D:Wn+1
i −Wni
∆t= − 1
∆x
(F i+1/2 −F i−1/2
)+ γ(Wn
i )(R(Wni )−Wn
i )
Limit:ρn+1i − ρni
∆t=bi+1/2∆x(ρni+1 − ρni )− bi−1/2∆x(ρni − ρni−1)
2∆x2
6−→ ∂tρ = div(p′(ρ)
κ∇ρ)
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 6/26
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Example of a non AP scheme in 1D
xi−1 xi xi+1
xi−1/2 xi+1/2xi−3/2 xi+3/2
xi−2 xi+2
∆x
Naïve scheme in 1D:Wn+1
i −Wni
∆t= − 1
∆x
(F i+1/2 −F i−1/2
)+ γ(Wn
i )(R(Wni )−Wn
i )
Limit:ρn+1i − ρni
∆t=bi+1/2∆x(ρni+1 − ρni )− bi−1/2∆x(ρni − ρni−1)
2∆x2
6−→ ∂tρ = div(p′(ρ)
κ∇ρ)
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 6/26
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Non-exhaustive state-of-the-art for AP schemes
1D meshes:1 control of numerical diffusion:
telegraph equations: Gosse and Toscani [GT02],M1 model: [BD06], [BC07], [BCD07], . . .Euler with gravity and friction: [CCGRS10],
2 ideas of hydrostatic reconstruction to have AP properties for the Eulermodel with friction: [BOP07],
3 using convergence speeds and finite differences: [ABN16],4 generalization of Gosse and Toscani: Berthon and Turpault [BT11].
2D unstructured meshes:1 MPFA based scheme: Buet, Després, and Franck [BDF12], with the Breil
and Maire scheme [BM07] as limit,2 SW with Manning-type friction: Duran, Marche, Turpault, and
Berthon [DMTB15].3 using the diamond scheme (Coudière, Vila, and Villedieu [CVV99]) for the
limit scheme: Berthon, Moebs, Sarazin-Desbois, and Turpault [BMST16],
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 7/26
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Non-exhaustive state-of-the-art for AP schemes
1D meshes:1 control of numerical diffusion:
telegraph equations: Gosse and Toscani [GT02],M1 model: [BD06], [BC07], [BCD07], . . .Euler with gravity and friction: [CCGRS10],
2 ideas of hydrostatic reconstruction to have AP properties for the Eulermodel with friction: [BOP07],
3 using convergence speeds and finite differences: [ABN16],4 generalization of Gosse and Toscani: Berthon and Turpault [BT11].
2D unstructured meshes:1 MPFA based scheme: Buet, Després, and Franck [BDF12], with the Breil
and Maire scheme [BM07] as limit,2 SW with Manning-type friction: Duran, Marche, Turpault, and
Berthon [DMTB15].3 using the diamond scheme (Coudière, Vila, and Villedieu [CVV99]) for the
limit scheme: Berthon, Moebs, Sarazin-Desbois, and Turpault [BMST16],
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 7/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV scheme
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Goal of the development
What we want:for any 2D unstructured mesh,for any system of conservation laws which could be written as (1),under a ‘hyperbolic’ CFL condition:
maxK,i
(bK,i
∆t
∆x
)≤ 1
2
stability,preservation of A,preservation of the asymptotic behaviour.
How to do it:choose a proper limit scheme for (2),build a global scheme which degenerates into it,
extension of existing TP flux,with a numerical diffusion correctly oriented.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 8/26
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Goal of the development
What we want:for any 2D unstructured mesh,for any system of conservation laws which could be written as (1),under a ‘hyperbolic’ CFL condition:
maxK,i
(bK,i
∆t
∆x
)≤ 1
2
stability,preservation of A,preservation of the asymptotic behaviour.
How to do it:choose a proper limit scheme for (2),build a global scheme which degenerates into it,
extension of existing TP flux,with a numerical diffusion correctly oriented.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 8/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV schemeChoice of a limit schemeHyperbolic partNumerical results for the hyperbolic partScheme for the full systemResults for the full system
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Choice of the limit scheme
FV scheme to discretize diffusion equations:
∂tw−div(f(w)∇w) = 0. (2)
Choice: scheme developed by Droniou and Le Potier in [DLP11]conservative and consistent with the diffusion equation on any mesh,satisfies the maximum principle and preserves A,nonlinear:
(f(wK)∇iwK) · nK,i '∑
J∈SK,i
νJK,i(w)(wJ − wK),
SK,i the set of points used for the reconstruction on edges i of cell K,νJK,i(w) ≥ 0.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 9/26
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Choice of the limit scheme
FV scheme to discretize diffusion equations:
∂tw−div(f(w)∇w) = 0. (2)
Choice: scheme developed by Droniou and Le Potier in [DLP11]conservative and consistent with the diffusion equation on any mesh,satisfies the maximum principle and preserves A,
nonlinear:
(f(wK)∇iwK) · nK,i '∑
J∈SK,i
νJK,i(w)(wJ − wK),
SK,i the set of points used for the reconstruction on edges i of cell K,νJK,i(w) ≥ 0.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 9/26
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Choice of the limit scheme
FV scheme to discretize diffusion equations:
∂tw−div(f(w)∇w) = 0. (2)
Choice: scheme developed by Droniou and Le Potier in [DLP11]conservative and consistent with the diffusion equation on any mesh,satisfies the maximum principle and preserves A,nonlinear:
(f(wK)∇iwK) · nK,i '∑
J∈SK,i
νJK,i(w)(wJ − wK),
SK,i the set of points used for the reconstruction on edges i of cell K,νJK,i(w) ≥ 0.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 9/26
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Quick presentation of the DLP scheme
K
L
J1
J2
nK,ii
nL,i
Two reconstructions:
∇iwK · nK,i =wMK,i
− wK
|KMK,i|
∇iwL · nL,i =wML,i
− wL
|LML,i|
Convex combination: θK,i + θL,i = 1, θK,i ≥ 0, θL,i ≥ 0
∇iwK · nK,i = θK,i(w)∇iwK · nK,i + θL,i(w)∇iwL · nL,i
=∑
J∈SK,i
νJK,i(w)(wJ − wK), with : νJK,i(w) ≥ 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 10/26
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Quick presentation of the DLP scheme
K
L
J1
J2
MK,i
ML,i
nK,ii
nL,i
Two reconstructions:
∇iwK · nK,i =wMK,i
− wK
|KMK,i|
∇iwL · nL,i =wML,i
− wL
|LML,i|
Convex combination: θK,i + θL,i = 1, θK,i ≥ 0, θL,i ≥ 0
∇iwK · nK,i = θK,i(w)∇iwK · nK,i + θL,i(w)∇iwL · nL,i
=∑
J∈SK,i
νJK,i(w)(wJ − wK), with : νJK,i(w) ≥ 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 10/26
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Quick presentation of the DLP scheme
K
L
J1
J2
MK,i
ML,i
nK,ii
nL,i
Two reconstructions:
∇iwK · nK,i =wMK,i
− wK
|KMK,i|
∇iwL · nL,i =wML,i
− wL
|LML,i|
Convex combination: θK,i + θL,i = 1, θK,i ≥ 0, θL,i ≥ 0
∇iwK · nK,i = θK,i(w)∇iwK · nK,i + θL,i(w)∇iwL · nL,i
=∑
J∈SK,i
νJK,i(w)(wJ − wK), with : νJK,i(w) ≥ 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 10/26
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Quick presentation of the DLP scheme
K
L
J1
J2
MK,i
ML,i
nK,ii
nL,i
Two reconstructions:
∇iwK · nK,i =wMK,i
− wK
|KMK,i|
∇iwL · nL,i =wML,i
− wL
|LML,i|
Convex combination: θK,i + θL,i = 1, θK,i ≥ 0, θL,i ≥ 0
∇iwK · nK,i = θK,i(w)∇iwK · nK,i + θL,i(w)∇iwL · nL,i
=∑
J∈SK,i
νJK,i(w)(wJ − wK), with : νJK,i(w) ≥ 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 10/26
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Quick presentation of the DLP scheme
K
L
J1
J2
MK,i
ML,i
nK,ii
nL,i
Two reconstructions:
∇iwK · nK,i =wMK,i
− wK
|KMK,i|
∇iwL · nL,i =wML,i
− wL
|LML,i|
Convex combination: θK,i + θL,i = 1, θK,i ≥ 0, θL,i ≥ 0
∇iwK · nK,i = θK,i(w)∇iwK · nK,i + θL,i(w)∇iwL · nL,i
=∑
J∈SK,i
νJK,i(w)(wJ − wK), with : νJK,i(w) ≥ 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 10/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV schemeChoice of a limit schemeHyperbolic partNumerical results for the hyperbolic partScheme for the full systemResults for the full system
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Scheme for the hyperbolic part
Wn+1K = Wn
K −∆t
|K|∑i∈EK
F i(WK ,WL, . . . ) · nK,i (3)
TheoremWe assume that the conservative and consistent flux F i has the followingproperties:
1 Combination: ∃ νJK,i ≥ 0, F i · nK,i =∑
J∈SK,i
νJK,iFKJ · ηKJ ,
2 Technical hypothesis:∑
i∈EK|ei|
∑J∈SK,i
νJK,i · ηKJ = 0.
Then the scheme (3) is stable, and preserves A under the classical followingCFL condition:
maxK∈M
J∈EK
(bKJ
∆t
δKJ
)≤ 1
2. (4)
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 11/26
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Scheme for the hyperbolic part
Wn+1K = Wn
K −∆t
|K|∑i∈EK
F i(WK ,WL, . . . ) · nK,i (3)
TheoremWe assume that the conservative and consistent flux F i has the followingproperties:
1 Combination: ∃ νJK,i ≥ 0, F i · nK,i =∑
J∈SK,i
νJK,iFKJ · ηKJ ,
2 Technical hypothesis:∑
i∈EK|ei|
∑J∈SK,i
νJK,i · ηKJ = 0.
Then the scheme (3) is stable, and preserves A under the classical followingCFL condition:
maxK∈M
J∈EK
(bKJ
∆t
δKJ
)≤ 1
2. (4)
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 11/26
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Example of fluxes (with Rusanov)
1 HLL-TP flux:
F i(WK ,WL) · nK,i =F(WK) + F(WL)
2· nK,i − bKL(WL −WK)
= FKL · nK,i
2 HLL-DLP flux:
F i(W) · nK,i =∑
J∈SK,i
νJK,i(W)FKJ · ηKJ
But. . .
1 HLL-TP flux:Fully respects the theorem,numerical diffusion oriented along KL.
2 HLL-DLP flux:Does not respect the technical hypothesis,right numerical diffusion.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 12/26
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Example of fluxes (with Rusanov)
1 HLL-TP flux:
F i(WK ,WL) · nK,i =F(WK) + F(WL)
2· nK,i − bKL(WL −WK)
= FKL · nK,i
2 HLL-DLP flux:
F i(W) · nK,i =∑
J∈SK,i
νJK,i(W)FKJ · ηKJ
But. . .
1 HLL-TP flux:Fully respects the theorem,numerical diffusion oriented along KL.
2 HLL-DLP flux:Does not respect the technical hypothesis,right numerical diffusion.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 12/26
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Example of fluxes (with Rusanov)
1 HLL-TP flux:
F i(WK ,WL) · nK,i =F(WK) + F(WL)
2· nK,i − bKL(WL −WK)
= FKL · nK,i
2 HLL-DLP flux:
F i(W) · nK,i =∑
J∈SK,i
νJK,i(W)FKJ · ηKJ
But. . .1 HLL-TP flux:
Fully respects the theorem,numerical diffusion oriented along KL.
2 HLL-DLP flux:Does not respect the technical hypothesis,right numerical diffusion.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 12/26
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Example of fluxes (with Rusanov)
1 HLL-TP flux:
F i(WK ,WL) · nK,i =F(WK) + F(WL)
2· nK,i − bKL(WL −WK)
= FKL · nK,i
2 HLL-DLP flux:
F i(W) · nK,i =∑
J∈SK,i
νJK,i(W)FKJ · ηKJ
But. . .1 HLL-TP flux:
Fully respects the theorem,numerical diffusion oriented along KL.
2 HLL-DLP flux:Does not respect the technical hypothesis,right numerical diffusion.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 12/26
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A posteriori procedure to preserve A
HLL-DLP flux W? W? ∈ A?Yes
NoHLL-TP flux
Wn Wn+1
1 W? is computed with the HLL-DLP flux and with the CFL condition (4),
2 Physical Admissiblility Detection (PAD):if W? ∈ A then the time iterations can continue,else, technical hypothesis 2 is enforced by using the HLL-TP flux on allnot-admissible cells.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 13/26
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A posteriori procedure to preserve A
HLL-DLP flux W? W? ∈ A?Yes
NoHLL-TP flux
Wn Wn+1
1 W? is computed with the HLL-DLP flux and with the CFL condition (4),2 Physical Admissiblility Detection (PAD):
if W? ∈ A then the time iterations can continue,else, technical hypothesis 2 is enforced by using the HLL-TP flux on allnot-admissible cells.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 13/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV schemeChoice of a limit schemeHyperbolic partNumerical results for the hyperbolic partScheme for the full systemResults for the full system
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Wind tunnel with step [WC84]
HLL-TP on 1.7× 103 cells
HLL-DLP on 1.7× 106 cellsTP correction < 1%
HLL-DLP on 1.7× 103 cells
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 14/26
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Wind tunnel with step [WC84]
HLL-TP on 1.7× 103 cells
HLL-DLP on 1.7× 106 cellsTP correction < 1%
HLL-DLP on 1.7× 103 cells
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 14/26
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2D Riemann problems with four shocks [KT02]
HLL-DLP 1.5× 105 cellsTP correction < 1% HLL-TP 1.5× 105 cells
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 15/26
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2D Riemann problems with four shocks [KT02]
HLL-DLP 1.5× 105 cellsTP correction < 1% HLL-TP 6× 105 cells
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 15/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV schemeChoice of a limit schemeHyperbolic partNumerical results for the hyperbolic partScheme for the full systemResults for the full system
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Scheme for the full system
Full system:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
Wn+1K = Wn
K −∆t
|K|∑i∈EK
|ei|FK,i · nK,i, (5)
Construction of FK,i with the technique of [BT11]:
FK,i · nK,i =∑
J∈SK,i
νJK,iFKJ · ηKJ
FKJ · ηKJ = αKJFKJ · ηKJ − (αKJ − αKK)F(WnK) · ηKJ
− (1− αKJ)bKJ(R(WnK)−Wn
K)
αKJ =bKJ
bKJ + γKδKJ∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 16/26
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Scheme for the full system
Full system:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
Wn+1K = Wn
K −∆t
|K|∑i∈EK
|ei|FK,i · nK,i, (5)
Construction of FK,i with the technique of [BT11]:
FK,i · nK,i =∑
J∈SK,i
νJK,iFKJ · ηKJ
FKJ · ηKJ = αKJFKJ · ηKJ − (αKJ − αKK)F(WnK) · ηKJ
− (1− αKJ)bKJ(R(WnK)−Wn
K)
αKJ =bKJ
bKJ + γKδKJ∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 16/26
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Scheme for the full system
Full system:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
Wn+1K = Wn
K −∆t
|K|∑i∈EK
|ei|FK,i · nK,i, (5)
Construction of FK,i with the technique of [BT11]:
FK,i · nK,i =∑
J∈SK,i
νJK,iFKJ · ηKJ
FKJ · ηKJ = αKJFKJ · ηKJ − (αKJ − αKK)F(WnK) · ηKJ
− (1− αKJ)bKJ(R(WnK)−Wn
K)
αKJ =bKJ
bKJ + γKδKJ∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 16/26
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Scheme for the full system
Full system:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
Wn+1K = Wn
K −∆t
|K|∑i∈EK
|ei|FK,i · nK,i, (5)
Construction of FK,i with the technique of [BT11]:
FK,i · nK,i =∑
J∈SK,i
νJK,iFKJ · ηKJ
FKJ · ηKJ = αKJFKJ · ηKJ − (αKJ − αKK)F(WnK) · ηKJ
− (1− αKJ)bKJ(R(WnK)−Wn
K)
αKJ =bKJ
bKJ + γKδKJ∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 16/26
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Scheme for the full system
Full system:
∂tW + div(F(W)) = γ(W)(R(W)−W) (1)
Wn+1K = Wn
K −∆t
|K|∑i∈EK
|ei|FK,i · nK,i, (5)
Construction of FK,i with the technique of [BT11]:
FK,i · nK,i =∑
J∈SK,i
νJK,iFKJ · ηKJ
FKJ · ηKJ = αKJFKJ · ηKJ − (αKJ − αKK)F(WnK) · ηKJ
− (1− αKJ)bKJ(R(WnK)−Wn
K)
αKJ =bKJ
bKJ + γKδKJ∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 16/26
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Theorem for the full scheme
Wn+1K = Wn
K −∆t
|K|∑i∈EK
|ei|FK,i · nK,i (5)
TheoremThe scheme (5) is consistent with the system of conservation laws (1), underthe same assumptions of the previous theorem. Moreover, it preserves the setof admissible states A under the CFL condition:
maxK∈M
J∈EK
(bKJ
∆t
δKJ
)≤ 1
2. (4)
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 17/26
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Scheme for the full model
Is the scheme with the source term AP?
: generally not: right direction for the numerical diffusion, but thecoefficient needs to be adjusted.
Equivalent formulation:
∂tW + div(F(W)) = γ(W)(R(W)−W), (1)= γ(W)(R(W)−W) + (γ − γ)W,
∂tW + div(F(W)) = (γ(W) + γ)(R(W)−W), (6)
with: γ(W) + γ > 0 and R(W) :=γR(W) + γW
γ + γ.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 18/26
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Scheme for the full model
Is the scheme with the source term AP?: generally not: right direction for the numerical diffusion, but thecoefficient needs to be adjusted.
Equivalent formulation:
∂tW + div(F(W)) = γ(W)(R(W)−W), (1)= γ(W)(R(W)−W) + (γ − γ)W,
∂tW + div(F(W)) = (γ(W) + γ)(R(W)−W), (6)
with: γ(W) + γ > 0 and R(W) :=γR(W) + γW
γ + γ.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 18/26
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Scheme for the full model
Is the scheme with the source term AP?: generally not: right direction for the numerical diffusion, but thecoefficient needs to be adjusted.
Equivalent formulation:
∂tW + div(F(W)) = γ(W)(R(W)−W), (1)= γ(W)(R(W)−W) + (γ − γ)W,
∂tW + div(F(W)) = (γ(W) + γ)(R(W)−W), (6)
with: γ(W) + γ > 0 and R(W) :=γR(W) + γW
γ + γ.
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 18/26
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Limit scheme for Euler, ∂tρ− div(p′(ρ)κ∇ρ
)= 0
Current limit scheme:
ρn+1K = ρnK +
∑i∈EK
∆t
|K| |ei|∑
J∈SK,i
νJK,i
b2KJ
2(κK + κJK,i)δKJ
(ρJ − ρK
).
AP correction κ:
νJK,i
b2KJ
2(κK + κJK,i)δKJ
= νJK,i
p′(ρ)iκ
(7)
Limit scheme with AP correction:
ρn+1K = ρnK +
∑i∈EK
∆t
|K| |ei|∑
J∈SK,i
νJK,i
p′(ρ)iκ
(ρJ − ρK)
−→ ∂tρ− div(p′(ρ)
κ∇ρ)
= 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 19/26
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Limit scheme for Euler, ∂tρ− div(p′(ρ)κ∇ρ
)= 0
Current limit scheme:
ρn+1K = ρnK +
∑i∈EK
∆t
|K| |ei|∑
J∈SK,i
νJK,i
b2KJ
2(κK + κJK,i)δKJ
(ρJ − ρK
).
AP correction κ:
νJK,i
b2KJ
2(κK + κJK,i)δKJ
= νJK,i
p′(ρ)iκ
(7)
Limit scheme with AP correction:
ρn+1K = ρnK +
∑i∈EK
∆t
|K| |ei|∑
J∈SK,i
νJK,i
p′(ρ)iκ
(ρJ − ρK)
−→ ∂tρ− div(p′(ρ)
κ∇ρ)
= 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 19/26
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Limit scheme for Euler, ∂tρ− div(p′(ρ)κ∇ρ
)= 0
Current limit scheme:
ρn+1K = ρnK +
∑i∈EK
∆t
|K| |ei|∑
J∈SK,i
νJK,i
b2KJ
2(κK + κJK,i)δKJ
(ρJ − ρK
).
AP correction κ:
νJK,i
b2KJ
2(κK + κJK,i)δKJ
= νJK,i
p′(ρ)iκ
(7)
Limit scheme with AP correction:
ρn+1K = ρnK +
∑i∈EK
∆t
|K| |ei|∑
J∈SK,i
νJK,i
p′(ρ)iκ
(ρJ − ρK)
−→ ∂tρ− div(p′(ρ)
κ∇ρ)
= 0
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 19/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV schemeChoice of a limit schemeHyperbolic partNumerical results for the hyperbolic partScheme for the full systemResults for the full system
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Comparison in the diffusion limit (pseudo 1D)
ρ0(x, y) = 0.1 exp((
x−0.50.01
)2)+ 0.1, u = 0, κ = 2000, tf = 10, 1.5× 103 cells
0 0.2 0.4 0.6 0.80.1
0.11
0.12
0.13
0.14
0.15
Position
Density
Parabolic 1DDLP
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 20/26
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Comparison in the diffusion limit (pseudo 1D)
ρ0(x, y) = 0.1 exp((
x−0.50.01
)2)+ 0.1, u = 0, κ = 2000, tf = 10, 1.5× 103 cells
0 0.2 0.4 0.6 0.80.1
0.11
0.12
0.13
0.14
0.15
Position
Density
Parabolic 1DDLPTP
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 20/26
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Comparison in the diffusion limit (pseudo 1D)
ρ0(x, y) = 0.1 exp((
x−0.50.01
)2)+ 0.1, u = 0, κ = 2000, tf = 10, 1.5× 103 cells
0 0.2 0.4 0.6 0.80.1
0.11
0.12
0.13
0.14
0.15
Position
Density
Parabolic 1DDLPTPNoAP
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 20/26
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Comparison in the diffusion limit (pseudo 1D)
ρ0(x, y) = 0.1 exp((
x−0.50.01
)2)+ 0.1, u = 0, κ = 2000, tf = 10, 1.5× 103 cells
0 0.2 0.4 0.6 0.80.1
0.11
0.12
0.13
0.14
0.15
Position
Density
Parabolic 1DDLPTPNoAPAP
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 20/26
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Comparison in the diffusion limit (2D)
ρ0(x, y) =
{1 if (x− 1
2)2
+ (y − 12
)2< 0.12
0.1 otherwise, u = 0, κ = 2000, tf = 10, 9.4× 103 cells
Figure: Mesh with privileged directionsF. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 21/26
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Comparison in the diffusion limit (2D)
(a) HLL-DLP-AP (b) DLP
(c) HLL-DLP-NoAP (d) HLL-TPF. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 22/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV scheme
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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High-order extension
Natural idea: MOOD : Clain, Diot, and Loubère [CDL11]
use a polynomial reconstruction of the solution W̃K(x),use the a posteriori limitation as in the TP flux correction.
However:
polynomial reconstruction need to be done by interface in the diffusivelimit: Clain, Machado, Nóbrega, and Pereira [CMNP13],α coefficients of Berthon and Turpault [BT11] : limit to first order.
New convex combination:
WK(x) = βKW̃K(x) + (1− βK)WK
βK =∆l
∆l + γKt∆xK∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 23/26
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High-order extension
Natural idea: MOOD : Clain, Diot, and Loubère [CDL11]
use a polynomial reconstruction of the solution W̃K(x),use the a posteriori limitation as in the TP flux correction.
However:polynomial reconstruction need to be done by interface in the diffusivelimit: Clain, Machado, Nóbrega, and Pereira [CMNP13],
α coefficients of Berthon and Turpault [BT11] : limit to first order.
New convex combination:
WK(x) = βKW̃K(x) + (1− βK)WK
βK =∆l
∆l + γKt∆xK∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 23/26
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High-order extension
Natural idea: MOOD : Clain, Diot, and Loubère [CDL11]
use a polynomial reconstruction of the solution W̃K(x),use the a posteriori limitation as in the TP flux correction.
However:polynomial reconstruction need to be done by interface in the diffusivelimit: Clain, Machado, Nóbrega, and Pereira [CMNP13],α coefficients of Berthon and Turpault [BT11] : limit to first order.
New convex combination:
WK(x) = βKW̃K(x) + (1− βK)WK
βK =∆l
∆l + γKt∆xK∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 23/26
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High-order extension
Natural idea: MOOD : Clain, Diot, and Loubère [CDL11]
use a polynomial reconstruction of the solution W̃K(x),use the a posteriori limitation as in the TP flux correction.
However:polynomial reconstruction need to be done by interface in the diffusivelimit: Clain, Machado, Nóbrega, and Pereira [CMNP13],α coefficients of Berthon and Turpault [BT11] : limit to first order.
New convex combination:
WK(x) = βKW̃K(x) + (1− βK)WK
βK =∆l
∆l + γKt∆xK∈ [0; 1]
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 23/26
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Cascade of schemes
High-order HLL-DLP-AP with W(x)
First-order HLL-DLP-AP
First-order HLL-TP
PrecisionStability,preservationof A
No activation when γt→∞
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 24/26
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Cascade of schemes
High-order HLL-DLP-AP with W(x)
First-order HLL-DLP-AP
First-order HLL-TP
PrecisionStability,preservationof A
No activation when γt→∞
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 24/26
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Cascade of schemes
High-order HLL-DLP-AP with W(x)
First-order HLL-DLP-AP
First-order HLL-TP
PrecisionStability,preservationof A
No activation when γt→∞
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 24/26
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Step with nonlinear friction: κ(ρ) = 10(ρ/7)3
HLL-DLP-AP-P0on 4× 104 cells
HLL-DLP-AP-P0on 1× 106 cells
HLL-DLP-AP-P1on 4× 104 cells
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Step with nonlinear friction: κ(ρ) = 10(ρ/7)3
HLL-DLP-AP-P0on 4× 104 cells
HLL-DLP-AP-P0on 1× 106 cells
HLL-DLP-AP-P1on 4× 104 cells
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 25/26
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Step with nonlinear friction: κ(ρ) = 10(ρ/7)3
HLL-DLP-AP-P0on 4× 104 cells
HLL-DLP-AP-P0on 1× 106 cells
HLL-DLP-AP-P1on 4× 104 cells
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 25/26
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Outline
1 General context and example
2 Development of an admissibility & asymptotic preserving FV scheme
3 High-order extension
4 Conclusion and perspectives
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim
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Conclusion and perspectives
Conclusiongeneric theory for various hyperbolic problems with asymptoticbehaviours,high-order scheme that preserve A and the asymptotic limit.
Perspectivesextend the limit scheme to take care of diffusion systems and morecomplex diffusion equation,full high-order scheme?
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Conclusion and perspectives
Conclusiongeneric theory for various hyperbolic problems with asymptoticbehaviours,high-order scheme that preserve A and the asymptotic limit.
Perspectivesextend the limit scheme to take care of diffusion systems and morecomplex diffusion equation,full high-order scheme?
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 26/26
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Thanks for your attention.
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References I
D. Aregba-Driollet, M. Briani, and R. Natalini, Time asymptotichigh order schemes for dissipative BGK hyperbolic systems,Numer. Math., vol. 132, no. 2, pp. 399–431, 2016 (pp. 17, 18).
C. Berthon, P. Charrier, and B. Dubroca, An HLLC scheme tosolve the M1 model of radiative transfer in two space dimensions,J. Sci. Comput., vol. 31, no. 3, pp. 347–389, 2007 (pp. 17, 18).
C. Berthon, P. G. LeFloch, and R. Turpault,Late-time/stiff-relaxation asymptotic-preserving approximationsof hyperbolic equations, Math. Comp., vol. 82, no. 282,pp. 831–860, 2013 (pp. 4, 5).
C. Berthon, G. Moebs, C. Sarazin-Desbois, and R. Turpault, Anasymptotic-preserving scheme for systems of conservation lawswith source terms on 2D unstructured meshes, Commun. Appl.Math. Comput. Sci., vol. 11, no. 1, pp. 55–77, 2016 (pp. 17, 18).
C. Berthon and R. Turpault, Asymptotic preserving HLLschemes, Numer. Methods Partial Differential Equations, vol.27, no. 6, pp. 1396–1422, 2011 (pp. 17, 18, 46–50, 66–69).
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References II
F. Bouchut, H. Ounaissa, and B. Perthame, Upwinding of thesource term at interfaces for euler equations with high friction,Comput. Math. Appl., vol. 53, no. 3-4, pp. 361–375, 2007(pp. 17, 18).
J. Breil and P.-H. Maire, A cell-centered diffusion scheme ontwo-dimensional unstructured meshes, J. Comput. Phys., vol.224, no. 2, pp. 785–823, 2007 (pp. 17, 18).
C. Buet and S. Cordier, An asymptotic preserving scheme forhydrodynamics radiative transfer models: Numerics for radiativetransfer, Numer. Math., vol. 108, no. 2, pp. 199–221, 2007(pp. 17, 18).
C. Buet and B. Després, Asymptotic preserving and positiveschemes for radiation hydrodynamics, J. Comput. Phys., vol.215, no. 2, pp. 717–740, 2006 (pp. 17, 18).
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References IIIC. Buet, B. Després, and E. Franck, Design of asymptoticpreserving finite volume schemes for the hyperbolic heatequation on unstructured meshes, Numer. Math., vol. 122, no.2, pp. 227–278, 2012 (pp. 17, 18).
C. Chalons, F. Coquel, E. Godlewski, P.-A. Raviart, andN. Seguin, Godunov-type schemes for hyperbolic systems withparameter-dependent source. The case of Euler system withfriction, Math. Models Methods Appl. Sci., vol. 20, no. 11,pp. 2109–2166, 2010 (pp. 17, 18).
S. Clain, S. Diot, and R. Loubère, A high-order finite volumemethod for systems of conservation laws—Multi-dimensionalOptimal Order Detection (MOOD), J. Comput. Phys., vol. 230,no. 10, pp. 4028–4050, 2011 (pp. 66–69).
S. Clain, G. J. Machado, J. M. Nóbrega, and R. M. S. Pereira, Asixth-order finite volume method for multidomainconvection-diffusion problem with discontinuous coefficients,Comput. Methods Appl. Mech. Engrg., vol. 267, pp. 43–64, 2013(pp. 66–69).
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 29/26
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References IV
Y. Coudière, J.-P. Vila, and P. Villedieu, Convergence rate of afinite volume scheme for a two-dimensional convection-diffusionproblem, M2AN Math. Model. Numer. Anal., vol. 33, no. 3,pp. 493–516, 1999 (pp. 17, 18).
J. Droniou and C. Le Potier, Construction and convergencestudy of schemes preserving the elliptic local maximumprinciple, SIAM J. Numer. Anal., vol. 49, no. 2, pp. 459–490,2011 (pp. 23–25).
A. Duran, F. Marche, R. Turpault, and C. Berthon, Asymptoticpreserving scheme for the shallow water equations with sourceterms on unstructured meshes, J. Comput. Phys., vol. 287,pp. 184–206, 2015 (pp. 17, 18).
L. Gosse and G. Toscani, An asymptotic-preservingwell-balanced scheme for the hyperbolic heat equations, C. R.Math. Acad. Sci. Paris, vol. 334, no. 4, pp. 337–342, 2002(pp. 17, 18).
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References V
A. Kurganov and E. Tadmor, Solution of two-dimensionalRiemann problems for gas dynamics without Riemann problemsolvers, Numer. Methods Partial Differential Equations, vol. 18,no. 5, pp. 584–608, 2002 (pp. 43, 44).
P. Woodward and P. Colella, The numerical simulation oftwo-dimensional fluid flow with strong shocks, J. Comput.Phys., vol. 54, no. 1, pp. 115–173, 1984 (pp. 41, 42).
F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 31/26