A high-order, admissibility and asymptotic-preserving ...€¦ · Outline 1 Generalcontextandexample 2 Developmentofanadmissibility&asymptoticpreservingFVscheme 3 High-orderextension

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





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]).


Problematic


∂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]).


Problematic


∂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.


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





A = {(ρ, ρu)T ∈ R3/ρ > 0}

Formalism of (1):

W = (ρ ρu)T

R(W) = (ρ 0)T


)Tγ(W) = κ > 0



κ∇ρ)

= 0





A = {(ρ, ρu)T ∈ R3/ρ > 0}

Formalism of (1):

W = (ρ ρu)T

R(W) = (ρ 0)T


)Tγ(W) = κ > 0



κ∇ρ)

= 0


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(w)∇w

)= 0

γt→∞

Numerical scheme



consistent?





(f(w)∇w

)= 0

γt→∞

Numerical scheme



consistent?





(f(w)∇w

)= 0

γt→∞

Numerical scheme



consistent?





(f(w)∇w

)= 0

γt→∞

Numerical scheme



consistent?


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)


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)


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],


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],


Outline






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.


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.


Outline


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




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.




∂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:


J∈SK,i






∂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:


J∈SK,i




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



K

L

J1

J2

MK,i

ML,i

nK,ii

nL,i



− wK

|KMK,i|


− wL

|LML,i|



=∑

J∈SK,i




K

L

J1

J2

MK,i

ML,i

nK,ii

nL,i



− wK

|KMK,i|


− wL

|LML,i|



=∑

J∈SK,i




K

L

J1

J2

MK,i

ML,i

nK,ii

nL,i



− wK

|KMK,i|


− wL

|LML,i|



=∑

J∈SK,i




K

L

J1

J2

MK,i

ML,i

nK,ii

nL,i



− wK

|KMK,i|


− wL

|LML,i|



=∑

J∈SK,i



Outline






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)


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)


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.



1 HLL-TP flux:



= FKL · nK,i

2 HLL-DLP flux:

F i(W) · nK,i =∑

J∈SK,i


But. . .

1 HLL-TP flux:Fully respects the theorem,numerical diffusion oriented along KL.




1 HLL-TP flux:



= FKL · nK,i

2 HLL-DLP flux:

F i(W) · nK,i =∑

J∈SK,i


But. . .1 HLL-TP flux:

Fully respects the theorem,numerical diffusion oriented along KL.




1 HLL-TP flux:



= FKL · nK,i

2 HLL-DLP flux:

F i(W) · nK,i =∑

J∈SK,i


But. . .1 HLL-TP flux:

Fully respects the theorem,numerical diffusion oriented along KL.



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.


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.


Outline






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


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


2D Riemann problems with four shocks [KT02]

HLL-DLP 1.5× 105 cellsTP correction < 1% HLL-TP 1.5× 105 cells


2D Riemann problems with four shocks [KT02]

HLL-DLP 1.5× 105 cellsTP correction < 1% HLL-TP 6× 105 cells


Outline






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]



Full system:

∂tW + div(F(W)) = γ(W)(R(W)−W) (1)

Wn+1K = Wn

K −∆t

|K|∑i∈EK



FK,i · nK,i =∑

J∈SK,i

νJK,iFKJ · ηKJ



K)

αKJ =bKJ




Full system:

∂tW + div(F(W)) = γ(W)(R(W)−W) (1)

Wn+1K = Wn

K −∆t

|K|∑i∈EK



FK,i · nK,i =∑

J∈SK,i

νJK,iFKJ · ηKJ



K)

αKJ =bKJ




Full system:

∂tW + div(F(W)) = γ(W)(R(W)−W) (1)

Wn+1K = Wn

K −∆t

|K|∑i∈EK



FK,i · nK,i =∑

J∈SK,i

νJK,iFKJ · ηKJ



K)

αKJ =bKJ




Full system:

∂tW + div(F(W)) = γ(W)(R(W)−W) (1)

Wn+1K = Wn

K −∆t

|K|∑i∈EK



FK,i · nK,i =∑

J∈SK,i

νJK,iFKJ · ηKJ



K)

αKJ =bKJ



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)


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.



Is the scheme with the source term AP?: generally not: right direction for the numerical diffusion, but thecoefficient needs to be adjusted.



∂tW + div(F(W)) = (γ(W) + γ)(R(W)−W), (6)


γ + γ.



Is the scheme with the source term AP?: generally not: right direction for the numerical diffusion, but thecoefficient needs to be adjusted.



∂tW + div(F(W)) = (γ(W) + γ)(R(W)−W), (6)


γ + γ.


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



)= 0


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)


ρn+1K = ρnK +

∑i∈EK

∆t

|K| |ei|∑

J∈SK,i

νJK,i

p′(ρ)iκ

(ρJ − ρK)


κ∇ρ)

= 0



)= 0


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)


ρn+1K = ρnK +

∑i∈EK

∆t

|K| |ei|∑

J∈SK,i

νJK,i

p′(ρ)iκ

(ρJ − ρK)


κ∇ρ)

= 0


Outline






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



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



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



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


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






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]





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.



βK =∆l

∆l + γKt∆xK∈ [0; 1]





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.



βK =∆l

∆l + γKt∆xK∈ [0; 1]





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.



βK =∆l

∆l + γKt∆xK∈ [0; 1]


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→∞


Cascade of schemes



First-order HLL-TP




Cascade of schemes



First-order HLL-TP




Step with nonlinear friction: κ(ρ) = 10(ρ/7)3

HLL-DLP-AP-P0on 4× 104 cells














Outline






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?


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?


Thanks for your attention.

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).


http://dx.doi.org/10.1007/s00211-015-0720-y

http://dx.doi.org/10.1007/s00211-015-0720-y

http://dx.doi.org/10.1007/s10915-006-9108-6

http://dx.doi.org/10.1007/s10915-006-9108-6

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A high-order, admissibility and asymptotic-preserving ...€¦ · Outline 1 Generalcontextandexample 2 Developmentofanadmissibility&asymptoticpreservingFVscheme 3 High-orderextension