**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).

**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**

**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**

**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 in

**x, with a direction**Ω. We consider the following equation :

∂tf(t,**x**,Ω) +Ω.∇f(t,**x**,Ω) =σ

S2

(f(t,**x**,Ω^{0})−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**) =

Ω

f(t,**x**,Ω)dΩand**F**(t,**x**) =

Ω

Ωf(t,**x**,Ω)dΩ.

### •

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

**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 in

**x, with a direction**Ω. We consider the following equation :

∂tf(t,**x**,Ω) +Ω.∇f(t,**x**,Ω) =σ

S2

(f(t,**x**,Ω^{0})−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**) =

Ω

f(t,**x**,Ω)dΩand**F**(t,**x**) =

Ω

Ωf(t,**x**,Ω)dΩ.

### •

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

**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}
ε^{div}** ^{F}**=0,

∂t**F**+ ^{1}
3ε∇E=−σ

ε^{2}** ^{F}**
.

### •

**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.

**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}
ε^{div}** ^{F}**=0,

∂t**F**+ ^{1}
3ε∇E=−σ

ε^{2}** ^{F}**
.

### •

**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.

**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**

**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:

Ej^{n}^{+}^{1}−Ej^{n}

∆t +
F_{j}_{+}1

2

−F_{j}_{−}1
2

ε∆x =0,
F^{n}^{+}^{1}

j+^{1}_{2}−F^{n}

j+^{1}_{2}

∆t +^{E}^{j}^{+}^{1}−Ej

ε∆x =−σ
ε^{2}^{F}^{j}^{+}^{1}2

.

### •

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.

**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:

Ej^{n}^{+}^{1}−Ej^{n}

∆t +
F_{j}_{+}1

2

−F_{j}_{−}1
2

ε∆x =0,
F^{n}^{+}^{1}

j+^{1}_{2}−F^{n}

j+^{1}_{2}

∆t +^{E}^{j}^{+}^{1}−Ej

ε∆x =−σ
ε^{2}^{F}^{j}^{+}^{1}2

.

### •

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.

**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:

Ej^{n}^{+}^{1}−Ej^{n}

∆t +
F_{j}_{+}1

2

−F_{j}_{−}1
2

ε∆x =0,
F^{n}^{+}^{1}

j+^{1}_{2}−F^{n}

j+^{1}_{2}

∆t +^{E}^{j}^{+}^{1}−Ej

ε∆x =−σ
ε^{2}^{F}^{j}^{+}^{1}2

.

### •

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.

**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

ε^{2}^{F},

=⇒∂tE−∂x

1 σ∂xE=0.

### •

Consistency error of the**upwind**scheme

• for the first equation :O ^{∆}_{ε}^{x}+∆t
,

• for the second equation :O

∆x2

ε +∆x+∆t .

### •

CFL condition :∆t1

∆xε+^{σ}

ε^{2}

≤1.

**Jin-Levermore scheme**

### •

Principle of design : we introduce the steady state∂xE=−^{σ}

_{ε}Fin the fluxes.

( E(xj) =E(x_{j}_{+}1
2

)−(xj−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

),
E(xj+1) =E(x_{j}_{+}1

2

)−(xj+1−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

). We obtain

E_{j}_{+}1

2

=_{Ej}_{+}_{Ej}_{+}_{1}

2 +^{Fj}^{−}^{Fj}_{2}^{+}^{1}
,
F_{j}_{+}1

2

=M
_{Fj}_{+}_{Fj}_{+}

1

2 +^{Ej}^{−}^{Ej}_{2}^{+}^{1}
.
withM= ^{2}^{ε}

2ε+σ∆x.

**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

ε^{2}^{F},

=⇒∂tE−∂x

1 σ∂xE=0.

### •

Consistency error of the**upwind**scheme

• for the first equation :O ^{∆}_{ε}^{x}+∆t
,

• for the second equation :O

∆x2

ε +∆x+∆t .

### •

CFL condition :∆t1

∆xε+^{σ}

ε^{2}

≤1.

**Jin-Levermore scheme**

### •

Principle of design : we introduce the steady state∂xE=−^{σ}

_{ε}Fin the fluxes.

### •

We write the relations( E(xj) =E(x_{j}_{+}1
2

) + (xj−x_{j}_{+}1
2

)∂xE(x_{j}_{+}1
2

),
E(xj+1) =E(x_{j}_{+}1

2

) + (xj+1−x_{j}_{+}1
2

)∂xE(x_{j}_{+}1
2

).

( E(xj) =E(x_{j}_{+}1
2

)−(xj−x_{j}_{+}1
2

)^{σ}

εF(x_{j}_{+}1
2

),
E(xj+1) =E(x_{j}_{+}1

2

)−(xj+1−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

). We obtain

E_{j}_{+}1

2

=_{Ej}_{+}_{Ej}_{+}

1

2 +^{Fj}^{−}^{Fj}_{2}^{+}^{1}
,
F_{j}_{+}1

2=M_{Fj}_{+}_{Fj}_{+}_{1}

2 +^{Ej}^{−}^{Ej}_{2}^{+}^{1}
.
withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

**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

ε^{2}^{F},

=⇒∂tE−∂x

1 σ∂xE=0.

### •

Consistency error of the**upwind**scheme

• for the first equation :O ^{∆}_{ε}^{x}+∆t
,

• for the second equation :O ∆x2

ε +∆x+∆t

.

### •

CFL condition :∆t1

∆xε+^{σ}

ε^{2}

≤1.

**Jin-Levermore scheme**

### •

Principle of design : we introduce the steady state∂xE=−^{σ}

_{ε}Fin the fluxes.

### •

We write the relations( E(xj) =E(x_{j}_{+}1
2

)−(xj−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

),
E(xj+1) =E(x_{j}_{+}1

2

)−(xj+1−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

).

We obtain

E_{j}_{+}1

2

=_{Ej}_{+}_{Ej}

+1

2 +^{Fj}^{−}^{Fj}_{2}^{+}^{1}
,
F_{j}_{+}1

2

=M_{Fj}_{+}_{Fj}_{+}_{1}

2 +^{Ej}^{−}^{Ej}_{2}^{+}^{1}
.
withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

**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

ε^{2}^{F},

=⇒∂tE−∂x

1 σ∂xE=0.

### •

Consistency error of the**upwind**scheme

• for the first equation :O ^{∆}_{ε}^{x}+∆t
,

• for the second equation :O

∆x2

ε +∆x+∆t .

### •

CFL condition :∆t1

∆xε+^{σ}

ε^{2}

≤1.

**Jin-Levermore scheme**

### •

Principle of design : we introduce the steady state∂xE=−^{σ}

_{ε}Fin the fluxes.

### •

We write the relations( E(xj) =E(x_{j}_{+}1
2

)−(xj−x_{j}_{+}1
2

)^{σ}

εF(x_{j}_{+}1
2

),
E(xj+1) =E(x_{j}_{+}1

2

)−(xj+1−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

).

Plugging the discrete equivalent of these relations in the fluxes.

We obtain

( Fj+Ej=F_{j}_{+}1
2

+E_{j}_{+}1
2

,
Fj+1−Ej+1=F_{j}_{+}1

2

−E_{j}_{+}1
2

.

E_{j}_{+}1

2

=_{Ej}_{+}_{Ej}_{+}

1

2 +^{Fj}^{−}^{Fj}_{2}^{+}^{1}
,
F_{j}_{+}1

2

=M
_{Fj}_{+}_{Fj}

+1

2 +^{Ej}^{−}^{Ej}_{2}^{+}^{1}
.
withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

**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

ε^{2}^{F},

=⇒∂tE−∂x

1 σ∂xE=0.

### •

Consistency error of the**upwind**scheme

• for the first equation :O ^{∆}_{ε}^{x}+∆t
,

• for the second equation :O

∆x2

ε +∆x+∆t .

### •

CFL condition :∆t1

∆xε+^{σ}

ε^{2}

≤1.

**Jin-Levermore scheme**

### •

Principle of design : we introduce the steady state∂xE=−^{σ}

_{ε}Fin the fluxes.

### •

We write the relations( E(xj) =E(x_{j}_{+}1
2

)−(xj−x_{j}_{+}1
2

)^{σ}

εF(x_{j}_{+}1
2

),
E(xj+1) =E(x_{j}_{+}1

2

)−(xj+1−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

).

We obtain

( Fj+Ej=F_{j}_{+}1
2

+E_{j}_{+}1
2

+^{σ∆}_{2}_{ε}^{x}F_{j}_{+}1
2

,
Fj+1−Ej+1=F_{j}_{+}1

2

−E_{j}_{+}1
2

+^{σ∆}_{2}_{ε}^{x}F_{j}_{+}1
2

.

E_{j}_{+}1

2

=_{Ej}_{+}_{Ej}_{+}

1

2 +^{Fj}^{−}^{Fj}_{2}^{+}^{1}
,
F_{j}_{+}1

2

=M
_{Fj}_{+}_{Fj}

+1

2 +^{Ej}^{−}^{Ej}_{2}^{+}^{1}
.
withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

**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

ε^{2}^{F},

=⇒∂tE−∂x

1 σ∂xE=0.

### •

Consistency error of the**upwind**scheme

• for the first equation :O ^{∆}_{ε}^{x}+∆t
,

• for the second equation :O

∆x2

ε +∆x+∆t .

### •

CFL condition :∆t1

∆xε+^{σ}

ε^{2}

≤1.

**Jin-Levermore scheme**

### •

Principle of design : we introduce the steady state∂xE=−^{σ}

_{ε}Fin the fluxes.

( E(xj) =E(x_{j}_{+}1
2

)−(xj−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

),
E(xj+1) =E(x_{j}_{+}1

2

)−(xj+1−x_{j}_{+}1
2

)^{σ}_{ε}F(x_{j}_{+}1
2

).

We obtain

E_{j}_{+}1

2

=_{Ej}_{+}_{Ej}_{+}_{1}

2 +^{Fj}^{−}^{Fj}_{2}^{+}^{1}
,
F_{j}_{+}1

2

=M
_{Fj}_{+}_{Fj}

+1

2 +^{Ej}^{−}^{Ej}_{2}^{+}^{1}
.
withM= ^{2}^{ε}

2ε+σ∆x.

**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 +M^{F n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{F n}_{x}^{j}^{−}^{1}−M^{E n}^{j}^{+}^{1}^{−}_{2}^{2E n}_{ε∆}^{j}_{x}^{+}^{E n}^{j}^{−}^{1}=0,

Fn+1 j −F nj

∆t +^{E n}^{j}^{+}_{2}^{1}_{ε∆}^{−}^{E n}_{x}^{j}^{−}^{1}−^{F n}^{j}^{+}^{1}^{−}_{2}^{2F n}_{ε∆}^{j}_{x}^{+}^{F n}^{j}^{−}^{1}+^{σ}

ε^{2}F_{j}^{n}=0.

(1)

withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

### •

Consistency error of Jin-Levermore :• for the first equation :O ∆x^{2}+ε∆x+∆t
,

• for the second equation :O
∆x^{2}

ε +∆x+∆t

.

### •

CFL condition of explicit scheme :∆t1

∆xε+^{σ}

ε^{2}

≤1.

### •

CFL condition of semi-implicit scheme :∆t_{∆}

^{1}

_{x}

_{ε}

≤1.

**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 +M^{F n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{F n}_{x}^{j}^{−}^{1}−M^{E n}^{j}^{+}^{1}^{−}_{2}^{2E n}_{ε∆}^{j}_{x}^{+}^{E n}^{j}^{−}^{1}=0,

Fn+1 j −F nj

∆t +^{E n}^{j}^{+}_{2}^{1}_{ε∆}^{−}^{E n}_{x}^{j}^{−}^{1}−^{F n}^{j}^{+}^{1}^{−}_{2}^{2F n}_{ε∆}^{j}_{x}^{+}^{F n}^{j}^{−}^{1}+^{σ}

ε^{2}F_{j}^{n}=0.

(1)

withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

### •

Consistency error of Jin-Levermore :• for the first equation :O ∆x^{2}+ε∆x+∆t
,

• for the second equation :O
∆x^{2}

ε +∆x+∆t

.

### •

CFL condition of explicit scheme :∆t1

∆xε+^{σ}

ε^{2}

≤1.

### •

CFL condition of semi-implicit scheme :∆t_{∆}

^{1}

_{x}

_{ε}

≤1.

**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 +M^{F n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{F n}_{x}^{j}^{−}^{1}−M^{E n}^{j}^{+}^{1}^{−}_{2}^{2E n}_{ε∆}^{j}_{x}^{+}^{E n}^{j}^{−}^{1}=0,

Fn+1 j −F nj

∆t +M^{E n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{E n}_{x}^{j}^{−}^{1}−M^{F n}^{j}^{+}^{1}^{−}_{2}^{2F n}_{ε∆}^{j}_{x}^{+}^{F n}^{j}^{−}^{1}+M^{σ}
ε^{2}F_{j}^{n}=0.

(2)

withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

### •

Consistency error of the**Gosse-Toscani**scheme :

• for the first equation :O ε∆x+∆x^{2}+∆t
,

• for the second equation :O(∆x+∆t).

### •

CFL condition of explicit scheme :∆t_{∆}

^{1}

_{x}

_{ε}

≤1.

### •

CFL condition of semi-implicit scheme :∆t1

∆xε+∆x2 σ

≤1.

### •

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

1

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

### •

**Remark**:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(

^{∆}

_{ε}

^{x}>>1).

**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

^{F n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{F n}_{x}^{j}^{−}^{1}−M^{E n}^{j}^{+}^{1}^{−}_{2}^{2E n}_{ε∆}^{j}_{x}^{+}^{E n}^{j}^{−}^{1}=0,

Fn+1 j −F nj

∆t +M^{E n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{E n}_{x}^{j}^{−}^{1}−M^{F n}^{j}^{+}^{1}^{−}_{2}^{2F n}_{ε∆}^{j}_{x}^{+}^{F n}^{j}^{−}^{1}+M^{σ}
ε^{2}F_{j}^{n}=0.

(2)

withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

### •

Consistency error of the**Gosse-Toscani**scheme :

• for the first equation :O ε∆x+∆x^{2}+∆t
,

• for the second equation :O(∆x+∆t).

### •

CFL condition of explicit scheme :∆t_{∆}

^{1}

_{x}

_{ε}

≤1.

### •

CFL condition of semi-implicit scheme :∆t1

∆xε+∆x2 σ

≤1.

### •

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

1

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

### •

**Remark**:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(

^{∆}

_{ε}

^{x}>>1).

**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

^{F n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{F n}_{x}^{j}^{−}^{1}−M^{E n}^{j}^{+}^{1}^{−}_{2}^{2E n}_{ε∆}^{j}_{x}^{+}^{E n}^{j}^{−}^{1}=0,

Fn+1 j −F nj

∆t +M^{E n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{E n}_{x}^{j}^{−}^{1}−M^{F n}^{j}^{+}^{1}^{−}_{2}^{2F n}_{ε∆}^{j}_{x}^{+}^{F n}^{j}^{−}^{1}+M^{σ}
ε^{2}F_{j}^{n}=0.

(2)

withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

### •

Consistency error of the**Gosse-Toscani**scheme :

• for the first equation :O ε∆x+∆x^{2}+∆t
,

• for the second equation :O(∆x+∆t).

### •

CFL condition of explicit scheme :∆t_{∆}

^{1}

_{x}

_{ε}

≤1.

### •

CFL condition of semi-implicit scheme :∆t1

∆xε+∆x2 σ

≤1.

### •

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

1

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

### •

**Remark**:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(

^{∆}

_{ε}

^{x}>>1).

**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

^{F n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{F n}_{x}^{j}^{−}^{1}−M^{E n}^{j}^{+}^{1}^{−}_{2}^{2E n}_{ε∆}^{j}_{x}^{+}^{E n}^{j}^{−}^{1}=0,

Fn+1 j −F nj

^{E n}^{j}^{+}^{1}_{2}_{ε∆}^{−}^{E n}_{x}^{j}^{−}^{1}−M^{F n}^{j}^{+}^{1}^{−}_{2}^{2F n}_{ε∆}^{j}_{x}^{+}^{F n}^{j}^{−}^{1}+M^{σ}
ε^{2}F_{j}^{n}=0.

(2)

withM=_{2}_{ε+σ∆}^{2}^{ε}_{x}.

### •

Consistency error of the**Gosse-Toscani**scheme :

• for the first equation :O ε∆x+∆x^{2}+∆t
,

• for the second equation :O(∆x+∆t).

### •

CFL condition of explicit scheme :∆t_{∆}

^{1}

_{x}

_{ε}

≤1.

### •

CFL condition of semi-implicit scheme :∆t1

∆xε+∆x2 σ

≤1.

### •

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

1

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

### •

**Remark**:For the two schemes, the numerical viscosity gives the diffusion limit scheme on coarse grids(

^{∆}

_{ε}

^{x}>>1).

**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 b}E|| ≤Ca,band||∂

_{ta}

_{,}

_{x b}F|| ≤εCa,b.

### •

The modified equation associated to the Upwind scheme is ∂tE+^{1}_{ε}∂xF−^{∆}_{2}^{x}_{ε}∂xxE=0,

∂tF+^{1}_{ε}∂xE−^{∆}_{2}_{ε}^{x}∂xxF=−^{σ}

ε^{2}F. ^{(3)}

### •

We plug the relationε∂xE+O(ε^{2}) =−σFin the first equation of (4), we obtain the diffusion limit

### •

The modified equation associated to the Gosse-Toscani scheme is ∂tE+M^{1}_{ε}∂xF−M^{∆}_{2}^{x}_{ε}∂xxE=0,

∂tF+M^{1}_{ε}∂xE−M^{∆}_{2}^{x}_{ε}∂xxF=−M^{σ}

ε^{2}F. ^{(4)}

### •

We plug the relationMε∂xE+O(ε^{2}) =−MσFin the first equation of (4)

∂tE−^{1}
σ∂xxE=0

### •

The scheme captures the diffusion limit (idem for the Jin-Levermore scheme).**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 b}E|| ≤Ca,band||∂

_{ta}

_{,}

_{x b}F|| ≤εCa,b.

### •

The modified equation associated to the Upwind scheme is ∂tE+^{1}

_{ε}∂xF−

^{∆}

_{2}

^{x}

_{ε}∂xxE=0,

∂tF+^{1}_{ε}∂xE−^{∆}_{2}_{ε}^{x}∂xxF=−^{σ}

ε^{2}F. ^{(3)}

### •

We plug the relationε∂xE+O(ε^{2}) =−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+M^{1}_{ε}∂xF−M^{∆}_{2}^{x}_{ε}∂xxE=0,

∂tF+M^{1}_{ε}∂xE−M^{∆}_{2}^{x}_{ε}∂xxF=−M^{σ}

ε^{2}F. ^{(4)}

### •

We plug the relationMε∂xE+O(ε^{2}) =−MσFin the first equation of (4)

∂tE−^{1}
σ∂xxE=0

### •

The scheme captures the diffusion limit (idem for the Jin-Levermore scheme).**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 b}E|| ≤Ca,band||∂

_{ta}

_{,}

_{x b}F|| ≤εCa,b.

### •

The modified equation associated to the Upwind scheme is ∂tE+^{1}

_{ε}∂xF−

^{∆}

_{2}

^{x}

_{ε}∂xxE=0,

∂tF+^{1}_{ε}∂xE−^{∆}_{2}_{ε}^{x}∂xxF=−^{σ}

ε^{2}F. ^{(3)}

### •

We plug the relationε∂xE+O(ε^{2}) =−σFin the first equation of (4), we obtain the diffusion limit

### •

On fine grid^{∆}

_{ε}

^{x}<<1, the diffusion limit is

∂tE−^{1}
σ∂xxE=0.

### •

The modified equation associated to the Gosse-Toscani scheme is ∂tE+M^{1}

_{ε}∂xF−M

^{∆}

_{2}

^{x}

_{ε}∂xxE=0,

∂tF+M^{1}_{ε}∂xE−M^{∆}_{2}^{x}_{ε}∂xxF=−M^{σ}

ε^{2}F. ^{(4)}

### •

We plug the relationMε∂xE+O(ε^{2}) =−MσFin the first equation of (4)

∂tE−^{1}
σ∂xxE=0