Seminary,IPP,Garching26june2012 EmmanuelFranck Cell-CenteredAsymptoticpreservingschemesforlineartransportonunstructuredmeshes

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Cell-Centered Asymptotic preserving schemes

for linear transport on unstructured meshes

Emmanuel Franck

CEA, DAM, DIF, F-91297 Arpajon, France - UPMC/LJLL

Seminary, IPP, Garching 26 june 2012

with : Christophe Buet (CEA), Bruno Després (UPMC).

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

1 Introduction

2 AP schemes in 1D and difficulties in 2D

3 AP schemes for hyperbolic heat equation in 2D

4 AP schemes for angular approximations of the transport equation

5 Numerical results

6 Conclusion

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Introduction

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Physical context and objectives

IFC simulations: interaction between the gas modelized by two-temperatures Euler equations and the radiation modelized by a linear transport equation.

Grey linear transport equation:f(x,Ω,t)≥0 the distribution function associated to the particles (photons or neutrons) located inx, with a directionΩ. We consider the following equation :

tf(t,x,Ω) +Ω.∇f(t,x,Ω) =σ

S2

(f(t,x,Ω0)−f(t,x,Ω))dΩ

0.

Diffusion limit: fort>>1 andσ>>1, the transport equation, tends toward the following diffusion equation

tE(t,x)−div 1

σ∇E(t,x)

=0,

withE(t,x) =

f(t,x,Ω)dΩandF(t,x) =

Ωf(t,x,Ω)dΩ.

Computation cost: The CPU cost is important, consequently one needs simplified models.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Physical context and objectives

IFC simulations: interaction between the gas modelized by two-temperatures Euler equations and the radiation modelized by a linear transport equation.

Grey linear transport equation:f(x,Ω,t)≥0 the distribution function associated to the particles (photons or neutrons) located inx, with a directionΩ. We consider the following equation :

tf(t,x,Ω) +Ω.∇f(t,x,Ω) =σ

S2

(f(t,x,Ω0)−f(t,x,Ω))dΩ0.

Diffusion limit: fort>>1 andσ>>1, the transport equation, tends toward the following diffusion equation

tE(t,x)−div 1

σ∇E(t,x)

=0,

withE(t,x) =

f(t,x,Ω)dΩandF(t,x) =

Ωf(t,x,Ω)dΩ.

Computation cost: The CPU cost is important, consequently one needs simplified models.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Approximation of transport equation

Simplified hyperbolic models, depend only on spaces variables.

Simplified models:

• Pnmodels : we develop the transport equation on the basis of spherical harmonics.

• Snmodels : we use a quadrature formula to discretize the collision operator.

• Mnmodels : non-linearPnmodels where the closure is obtained by minimizing the entropy.

P1model :





tE+1 εdivF=0,

tF+ 1 3ε∇E=−σ

ε2F .

Adapted numerical methods: asymptotic preserving (AP) finite volume schemes capturing the diffusion limit.

Aims :

Design of cell-centered finite volume schemes for the simplified models capturing the diffusion limit on unstructured meshes.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Approximation of transport equation

Simplified hyperbolic models, depend only on spaces variables.

Simplified models:

• Pnmodels : we develop the transport equation on the basis of spherical harmonics.

• Snmodels : we use a quadrature formula to discretize the collision operator.

• Mnmodels : non-linearPnmodels where the closure is obtained by minimizing the entropy.

P1model :





tE+1 εdivF=0,

tF+ 1 3ε∇E=−σ

ε2F .

Adapted numerical methods: asymptotic preserving (AP) finite volume schemes capturing the diffusion limit.

Aims :

Design of cell-centered finite volume schemes for the simplified models capturing the diffusion limit on unstructured meshes.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes in 1D and difficulties in 2D

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Staggered and centered schemes

Contrary to the Godunov schemes (HLL, Rusanov, upwind)the centered scheme for the hyperbolic heat equation is AP.

The limit diffusion scheme admit spurious modes.

The centered scheme is not stable.

The staggered schemeis also asymptotic preserving:





Ejn+1−Ejn

∆t + Fj+1

2

−Fj1 2

ε∆x =0, Fn+1

j+12−Fn

j+12

∆t +Ej+1−Ej

ε∆x =−σ ε2Fj+12

.

But the staggered scheme does not preserve the maximum principleE+F>0,E−F>0 in the transport regime.

CEA Constraints

• Design of schemes which are equal to the upwind scheme whenσ=0 to preserve the transport properties (maximum principle, entropy etc).

• Design of cell-centered schemes. Indeed the hydrodynamic and diffusion codes are coupled with the transport problem use cell-centered methods.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Staggered and centered schemes

Contrary to the Godunov schemes (HLL, Rusanov, upwind)the centered scheme for the hyperbolic heat equation is AP.

The limit diffusion scheme admit spurious modes.

The centered scheme is not stable.

The staggered schemeis also asymptotic preserving:





Ejn+1−Ejn

∆t + Fj+1

2

−Fj1 2

ε∆x =0, Fn+1

j+12−Fn

j+12

∆t +Ej+1−Ej

ε∆x =−σ ε2Fj+12

.

But the staggered scheme does not preserve the maximum principleE+F>0,E−F>0 in the transport regime.

CEA Constraints

• Design of schemes which are equal to the upwind scheme whenσ=0 to preserve the transport properties (maximum principle, entropy etc).

• Design of cell-centered schemes. Indeed the hydrodynamic and diffusion codes are coupled with the transport problem use cell-centered methods.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Staggered and centered schemes

Contrary to the Godunov schemes (HLL, Rusanov, upwind)the centered scheme for the hyperbolic heat equation is AP.

The limit diffusion scheme admit spurious modes.

The centered scheme is not stable.

The staggered schemeis also asymptotic preserving:





Ejn+1−Ejn

∆t + Fj+1

2

−Fj1 2

ε∆x =0, Fn+1

j+12−Fn

j+12

∆t +Ej+1−Ej

ε∆x =−σ ε2Fj+12

.

But the staggered scheme does not preserve the maximum principleE+F>0,E−F>0 in the transport regime.

CEA Constraints

• Design of schemes which are equal to the upwind scheme whenσ=0 to preserve the transport properties (maximum principle, entropy etc).

• Design of cell-centered schemes. Indeed the hydrodynamic and diffusion codes are coupled with the transport problem use cell-centered methods.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes : design and examples

Hyperbolic heat equation :





tE+1 ε∂xF=0,

tF+1 ε∂xE=−σ

ε2F,

=⇒∂tE−∂x

1 σ∂xE=0.

Consistency error of theupwindscheme

• for the first equation :O εx+∆t ,

• for the second equation :O

x2

ε +∆x+∆t .

CFL condition :∆t

1

xε+σ

ε2

≤1.

Jin-Levermore scheme

Principle of design : we introduce the steady state∂xE=−σεFin the fluxes.

( E(xj) =E(xj+1 2

)−(xj−xj+1 2

)σεF(xj+1 2

), E(xj+1) =E(xj+1

2

)−(xj+1−xj+1 2

)σεF(xj+1 2

). We obtain

 Ej+1

2

=Ej+Ej+1

2 +FjFj2+1 , Fj+1

2

=M Fj+Fj+

1

2 +EjEj2+1 . withM= 2ε

2ε+σ∆x.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes : design and examples

Hyperbolic heat equation :





tE+1 ε∂xF=0,

tF+1 ε∂xE=−σ

ε2F,

=⇒∂tE−∂x

1 σ∂xE=0.

Consistency error of theupwindscheme

• for the first equation :O εx+∆t ,

• for the second equation :O

x2

ε +∆x+∆t .

CFL condition :∆t

1

xε+σ

ε2

≤1.

Jin-Levermore scheme

Principle of design : we introduce the steady state∂xE=−σεFin the fluxes.

We write the relations

( E(xj) =E(xj+1 2

) + (xj−xj+1 2

)∂xE(xj+1 2

), E(xj+1) =E(xj+1

2

) + (xj+1−xj+1 2

)∂xE(xj+1 2

).

( E(xj) =E(xj+1 2

)−(xj−xj+1 2

)σ

εF(xj+1 2

), E(xj+1) =E(xj+1

2

)−(xj+1−xj+1 2

)σεF(xj+1 2

). We obtain

 Ej+1

2

=Ej+Ej+

1

2 +FjFj2+1 , Fj+1

2=MFj+Fj+1

2 +EjEj2+1 . withM=2ε+σ∆2εx.

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Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes : design and examples

Hyperbolic heat equation :





tE+1 ε∂xF=0,

tF+1 ε∂xE=−σ

ε2F,

=⇒∂tE−∂x

1 σ∂xE=0.

Consistency error of theupwindscheme

• for the first equation :O εx+∆t ,

• for the second equation :O x2

ε +∆x+∆t

.

CFL condition :∆t

1

xε+σ

ε2

≤1.

Jin-Levermore scheme

Principle of design : we introduce the steady state∂xE=−σεFin the fluxes.

We write the relations

( E(xj) =E(xj+1 2

)−(xj−xj+1 2

)σεF(xj+1 2

), E(xj+1) =E(xj+1

2

)−(xj+1−xj+1 2

)σεF(xj+1 2

).

We obtain

 Ej+1

2

=Ej+Ej

+1

2 +FjFj2+1 , Fj+1

2

=MFj+Fj+1

2 +EjEj2+1 . withM=2ε+σ∆2εx.

(15)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes : design and examples

Hyperbolic heat equation :





tE+1 ε∂xF=0,

tF+1 ε∂xE=−σ

ε2F,

=⇒∂tE−∂x

1 σ∂xE=0.

Consistency error of theupwindscheme

• for the first equation :O εx+∆t ,

• for the second equation :O

x2

ε +∆x+∆t .

CFL condition :∆t

1

xε+σ

ε2

≤1.

Jin-Levermore scheme

Principle of design : we introduce the steady state∂xE=−σεFin the fluxes.

We write the relations

( E(xj) =E(xj+1 2

)−(xj−xj+1 2

)σ

εF(xj+1 2

), E(xj+1) =E(xj+1

2

)−(xj+1−xj+1 2

)σεF(xj+1 2

).

Plugging the discrete equivalent of these relations in the fluxes.

We obtain

( Fj+Ej=Fj+1 2

+Ej+1 2

, Fj+1−Ej+1=Fj+1

2

−Ej+1 2

.

 Ej+1

2

=Ej+Ej+

1

2 +FjFj2+1 , Fj+1

2

=M Fj+Fj

+1

2 +EjEj2+1 . withM=2ε+σ∆2εx.

(16)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes : design and examples

Hyperbolic heat equation :





tE+1 ε∂xF=0,

tF+1 ε∂xE=−σ

ε2F,

=⇒∂tE−∂x

1 σ∂xE=0.

Consistency error of theupwindscheme

• for the first equation :O εx+∆t ,

• for the second equation :O

x2

ε +∆x+∆t .

CFL condition :∆t

1

xε+σ

ε2

≤1.

Jin-Levermore scheme

Principle of design : we introduce the steady state∂xE=−σεFin the fluxes.

We write the relations

( E(xj) =E(xj+1 2

)−(xj−xj+1 2

)σ

εF(xj+1 2

), E(xj+1) =E(xj+1

2

)−(xj+1−xj+1 2

)σεF(xj+1 2

).

We obtain

( Fj+Ej=Fj+1 2

+Ej+1 2

+σ∆2εxFj+1 2

, Fj+1−Ej+1=Fj+1

2

−Ej+1 2

+σ∆2εxFj+1 2

.

 Ej+1

2

=Ej+Ej+

1

2 +FjFj2+1 , Fj+1

2

=M Fj+Fj

+1

2 +EjEj2+1 . withM=2ε+σ∆2εx.

(17)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes : design and examples

Hyperbolic heat equation :





tE+1 ε∂xF=0,

tF+1 ε∂xE=−σ

ε2F,

=⇒∂tE−∂x

1 σ∂xE=0.

Consistency error of theupwindscheme

• for the first equation :O εx+∆t ,

• for the second equation :O

x2

ε +∆x+∆t .

CFL condition :∆t

1

xε+σ

ε2

≤1.

Jin-Levermore scheme

Principle of design : we introduce the steady state∂xE=−σεFin the fluxes.

( E(xj) =E(xj+1 2

)−(xj−xj+1 2

)σεF(xj+1 2

), E(xj+1) =E(xj+1

2

)−(xj+1−xj+1 2

)σεF(xj+1 2

).

We obtain

 Ej+1

2

=Ej+Ej+1

2 +FjFj2+1 , Fj+1

2

=M Fj+Fj

+1

2 +EjEj2+1 . withM= 2ε

2ε+σ∆x.

(18)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes in 1D

The Jin-Levermore scheme is

En+1

j E n

j

t +MF nj+12ε∆F nxj1−ME nj+122E nε∆jx+E nj1=0,

Fn+1 j F nj

t +E nj+21ε∆E nxj1F nj+122F nε∆jx+F nj1+σ

ε2Fjn=0.

(1)

withM=2ε+σ∆2εx.

Consistency error of Jin-Levermore :

• for the first equation :O ∆x2+ε∆x+∆t ,

• for the second equation :O ∆x2

ε +∆x+∆t

.

CFL condition of explicit scheme :∆t

1

xε+σ

ε2

≤1.

CFL condition of semi-implicit scheme :∆t 1xε

≤1.

(19)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes in 1D

The Jin-Levermore scheme is

En+1

j E n

j

t +MF nj+12ε∆F nxj1−ME nj+122E nε∆jx+E nj1=0,

Fn+1 j F nj

t +E nj+21ε∆E nxj1F nj+122F nε∆jx+F nj1+σ

ε2Fjn=0.

(1)

withM=2ε+σ∆2εx.

Consistency error of Jin-Levermore :

• for the first equation :O ∆x2+ε∆x+∆t ,

• for the second equation :O ∆x2

ε +∆x+∆t

.

CFL condition of explicit scheme :∆t

1

xε+σ

ε2

≤1.

CFL condition of semi-implicit scheme :∆t 1xε

≤1.

(20)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes in 1D

Gosse-Toscani scheme

Principle of design : localization of source terms at the interfaces, which induces a stationary wave in the Riemann problem.

En+1

j E n

j

t +MF nj+12ε∆F nxj1−ME nj+122E nε∆jx+E nj1=0,

Fn+1 j F nj

t +ME nj+12ε∆E nxj1−MF nj+122F nε∆jx+F nj1+Mσ ε2Fjn=0.

(2)

withM=2ε+σ∆2εx.

Consistency error of theGosse-Toscanischeme :

• for the first equation :O ε∆x+∆x2+∆t ,

• for the second equation :O(∆x+∆t).

CFL condition of explicit scheme :∆t 1xε

≤1.

CFL condition of semi-implicit scheme :∆t

1

xε+x2 σ

≤1.

Remark: The Jin-Levermore scheme (1) with the discretization of the source term

1

2(Fj+1/2+Fj1/2)is equal to the Gosse-Toscani scheme.

Remark:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(εx>>1).

(21)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes in 1D

Gosse-Toscani scheme

Principle of design : localization of source terms at the interfaces, which induces a stationary wave in the Riemann problem.

En+1

j E n

j

t +MF nj+12ε∆F nxj1−ME nj+122E nε∆jx+E nj1=0,

Fn+1 j F nj

t +ME nj+12ε∆E nxj1−MF nj+122F nε∆jx+F nj1+Mσ ε2Fjn=0.

(2)

withM=2ε+σ∆2εx.

Consistency error of theGosse-Toscanischeme :

• for the first equation :O ε∆x+∆x2+∆t ,

• for the second equation :O(∆x+∆t).

CFL condition of explicit scheme :∆t 1xε

≤1.

CFL condition of semi-implicit scheme :∆t

1

xε+x2 σ

≤1.

Remark: The Jin-Levermore scheme (1) with the discretization of the source term

1

2(Fj+1/2+Fj1/2)is equal to the Gosse-Toscani scheme.

Remark:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(εx>>1).

(22)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes in 1D

Gosse-Toscani scheme

Principle of design : localization of source terms at the interfaces, which induces a stationary wave in the Riemann problem.

En+1

j E n

j

t +MF nj+12ε∆F nxj1−ME nj+122E nε∆jx+E nj1=0,

Fn+1 j F nj

t +ME nj+12ε∆E nxj1−MF nj+122F nε∆jx+F nj1+Mσ ε2Fjn=0.

(2)

withM=2ε+σ∆2εx.

Consistency error of theGosse-Toscanischeme :

• for the first equation :O ε∆x+∆x2+∆t ,

• for the second equation :O(∆x+∆t).

CFL condition of explicit scheme :∆t 1xε

≤1.

CFL condition of semi-implicit scheme :∆t

1

xε+x2 σ

≤1.

Remark: The Jin-Levermore scheme (1) with the discretization of the source term

1

2(Fj+1/2+Fj1/2)is equal to the Gosse-Toscani scheme.

Remark:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(εx>>1).

(23)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

AP schemes in 1D

Gosse-Toscani scheme

Principle of design : localization of source terms at the interfaces, which induces a stationary wave in the Riemann problem.

En+1

j E n

j

t +MF nj+12ε∆F nxj1−ME nj+122E nε∆jx+E nj1=0,

Fn+1 j F nj

t +ME nj+12ε∆E nxj1−MF nj+122F nε∆jx+F nj1+Mσ ε2Fjn=0.

(2)

withM=2ε+σ∆2εx.

Consistency error of theGosse-Toscanischeme :

• for the first equation :O ε∆x+∆x2+∆t ,

• for the second equation :O(∆x+∆t).

CFL condition of explicit scheme :∆t 1xε

≤1.

CFL condition of semi-implicit scheme :∆t

1

xε+x2 σ

≤1.

Remark: The Jin-Levermore scheme (1) with the discretization of the source term

1

2(Fj+1/2+Fj1/2)is equal to the Gosse-Toscani scheme.

Remark:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(εx>>1).

(24)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Analysis of AP schemes : modified equations

To understand the behaviour of the scheme, we use the modified equations method.

We assume that||∂ta,x bE|| ≤Ca,band||∂ta,x bF|| ≤εCa,b.

The modified equation associated to the Upwind scheme is

tE+1εxF−2xεxxE=0,

tF+1εxE−2εxxxF=−σ

ε2F. (3)

We plug the relationε∂xE+O(ε2) =−σFin the first equation of (4), we obtain the diffusion limit

The modified equation associated to the Gosse-Toscani scheme is

tE+M1εxF−M2xεxxE=0,

tF+M1εxE−M2xεxxF=−Mσ

ε2F. (4)

We plug the relationMε∂xE+O(ε2) =−MσFin the first equation of (4)

tE−1 σ∂xxE=0

The scheme captures the diffusion limit (idem for the Jin-Levermore scheme).

(25)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Analysis of AP schemes : modified equations

To understand the behaviour of the scheme, we use the modified equations method.

We assume that||∂ta,x bE|| ≤Ca,band||∂ta,x bF|| ≤εCa,b.

The modified equation associated to the Upwind scheme is ∂tE+1εxF−2xεxxE=0,

tF+1εxE−2εxxxF=−σ

ε2F. (3)

We plug the relationε∂xE+O(ε2) =−σFin the first equation of (4), we obtain the diffusion limit

tE−1 σ∂xxE−∆x

2ε∂xxE=0.

The modified equation associated to the Gosse-Toscani scheme is

tE+M1εxF−M2xεxxE=0,

tF+M1εxE−M2xεxxF=−Mσ

ε2F. (4)

We plug the relationMε∂xE+O(ε2) =−MσFin the first equation of (4)

tE−1 σ∂xxE=0

The scheme captures the diffusion limit (idem for the Jin-Levermore scheme).

(26)

Introduction AP schemes in 1D and difficulties in 2D AP schemes for hyperbolic heat equation in 2D AP schemes for angular approximations of the transport equation Numerical results Conclusion

Analysis of AP schemes : modified equations

To understand the behaviour of the scheme, we use the modified equations method.

We assume that||∂ta,x bE|| ≤Ca,band||∂ta,x bF|| ≤εCa,b.

The modified equation associated to the Upwind scheme is ∂tE+1εxF−2xεxxE=0,

tF+1εxE−2εxxxF=−σ

ε2F. (3)

We plug the relationε∂xE+O(ε2) =−σFin the first equation of (4), we obtain the diffusion limit

On fine gridεx<<1, the diffusion limit is

tE−1 σ∂xxE=0.

The modified equation associated to the Gosse-Toscani scheme is ∂tE+M1εxF−M2xεxxE=0,

tF+M1εxE−M2xεxxF=−Mσ

ε2F. (4)

We plug the relationMε∂xE+O(ε2) =−MσFin the first equation of (4)

tE−1 σ∂xxE=0

The scheme captures the diffusion limit (idem for the Jin-Levermore scheme).

Figure

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Références

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