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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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thanks for your attention.

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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).

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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).

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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).

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References V

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F. Blachère (Nantes) SHARK-FV, May 2016, Póvoa de Varzim 31/26

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