## Design and analysis of cell-centered finite volume schemes in the diffusion limit on distorted meshes

### C. Buet

^{1}

### , B. Despr´ es

^{2}

### , E. Franck

^{3}

### , T. Leroy

^{4}

### NumHyp2013, Aachen, 26 September 2013

1CEA, DAM, DIF, France

2LJLL, UMPC, Paris, France

3Max-Planck-Institut f¨ur Plasmaphysik, Garching, Germany

4CEA, DAM, DIF and LJLL , UPMC, Paris, France

## Outline

1 Physical and mathematical context

2 AP schemes on unstructured meshes for theP1model

3 Nonlinear extension: M1and Euler models

### Physical and mathematical context

## Stiff hyperbolic systems

Hyperbolic systems with stiff source terms:

∂tU+1

ε∂xF(U) +1

ε∂yG(U) =−σ

ε^{2}R(U),U∈R^{n}
withε∈]0,1] etσ >0.

Diffusion limitforε→0:

∂tV−div (K(∇V, σ)) = 0, V∈KerR.

Applications: biology, neutronic, fluids dynamic, plasma physic,Radiative Hydrodynamics for IFC(Hydrodynamic + linear transport (talk of C. Hauck)).

## Asymptotic preserving schemes

P1Model:

∂tE+^{1}_{ε}∂xF= 0,

∂tF+^{1}_{ε}∂xE=−^{σ}

ε^{2}F, −→∂tE−∂x

1 σ∂xE

= 0. (1)

ε→0

### P

_{h}

^{0}

### P

^{ε}

h_{→}0

### P

^{0}

ε→0
h_{→}0

### P

_{h}

^{ε}

### Figure:

The AP diagramConsistency ofGodunov-type
schemes:O(^{∆x}_{ε} + ∆t).

CFL condition : ∆t(_{∆xε}^{1} + ^{σ}

ε^{2})≤1.

Consistency of AP schemes:

O(∆x+ ∆t).

CFL condition: ∆t≤∆x^{2}+ε∆x
AP vs non AP schemes: Very
important reduction of CPU cost.

Godunov-type AP schemes are obtainedplugging the source terms in the fluxes(S.

Jin-D. Levermore, L.Gosse and al, C. Berthon, R. Turpault and al ... ).

The problem of consistency for the Godunov-type schemes come from to thenumerical
viscosity homogeneous toO(^{∆x}_{ε} ).

## Why unstructured meshes ?

Applications: coupling between radiation and hydrodynamic.

Some hydrodynamic codes:

multi-material Langrangian or ALE cell-centered schemes.

Example of mesh obtained with ALE code (right).

Aim:Design and analyze cell-centered AP schemes for linear transport on general meshes.

−1.5×10^{−18} 1.0×10^{0}

0 1

−1.5×10^{−18} 1.0×10^{0}

0 1

## 2D Asymptotic preserving schemes

2D Classical extension Jin-Levermore scheme: modify the upwind fluxes (1D fluxes write in the normal direction) plugging the steady states into the fluxes (JL method).

**x***j*
**x***r+1*

**x***r−1*
*l**jk*

**x***r*

CellΩ*j*

CellΩ*k*
**x***k*
**n***jk*

ljk andnjk are the lenght and the normal associated to the edge∂Ωjk.

Asymptotic limit of AP scheme: |Ω_{j}|∂_{t}E_{j}(t)−1
σ

X k

l_{jk}E_{k}^{n}−E_{j}^{n}
d(x_{j},x_{k})= 0.

||P_{h}^{0}−Ph|| →0 only under very strong geometrical conditions.

Non convergenceof 2D AP schemes on general meshes∀ε.

## Examples of unstructured meshes

Random mesh Smooth mesh

0 0.5 1 1.5 2

0 0.5 1 1.5 2

0 0.5 1 1.5 2

0 0.5 1 1.5 2

Ramdom trig mesh Kershaw mesh

### AP schemes on unstructured meshes for the P

1### model

## 2D AP schemes: Principle and notations

Idea:Nodalformulation of finite volume methods (fluxes localized at the node) for the P1model + the Jin-Levermore method.

Geometrical quantities Control volume

**x***j*
**x***r+1*

**x***r−1*
**x***r*

CellΩ*j*

CellΩ*k*
*l**jr***n***jr*

**x***j*
**x**1
2

**x***r*

**x***j*−^{1}2

*V**r*

Geometrical quantities defined byljrnjr =∇xr|Ωj|(Left).

P

jl_{jr}n_{jr}=P

rl_{jr}n_{jr}=0.

Vr control volume (right).

## 2D AP schemes

Nodal AP schemes:

|Ω_{j}|∂tE_{j}(t) +^{1}_{ε}P

rl_{jr}(Fr,n_{jr}) = 0,

|Ωj|∂tFj(t) +^{1}_{ε}P

rljrEnjr=Sj. Classical nodal fluxes:

En_{jr}−l_{jr}E_{j}n_{jr}=αb_{jr}(F_{j}−Fr),
P

jEnjr=0, withαbjr=ljrnjr⊗njr.

New fluxes obtained plugging the steady state∇E=−^{σ}_{ε}F:
( Enjr−ljrEjnjr=bαjr(Fj−Fr)−^{σ}

εβbjrFr, P

jαbjr+^{σ}_{ε}P

jβbjr

Fr=P

jljrEjnjr+P

jαbjrFj. withβbjr=ljrnjr⊗(xr−xj).

Source term: (1)Sj=−^{σ}

ε^{2} |Ωj|Fj, (2)Sj=−^{σ}

ε^{2}

P

rβbjrFr, P

rβbjr= ˆId|Ω_{j}|.

## Time discretization for AP schemes

Formulation of the scheme with the source term (2) and local semi-implicit scheme:

|Ωj|E_{j}^{n+1}−E_{j}^{n}
4t +1

ε X

r

ljr(MrFr,njr) = 0,

|Ωj|F^{n+1}_{j} −F^{n}_{j}
4t +1

ε X

r

Enjr=−1 ε

X

r

αbjr(bId−Mr)

!
F^{n+1}_{j} .

with (

Enjr−ljrEjnjr=αbjrMr(Fj−Fr), P

jαbjr

Fr=P

jljrEjnjr+P

jαbjrFj.

Mr=

X

j

αbjr+σ ε

X

j

βbjr

−1

X

j

αbjr

The semi-implicit scheme is stable on a CFL condition independent toε(numerically).

The implicite scheme is stable.

## Assumptions for the proof

Geometrical assumptions (u,

P

jljrnjr⊗(xr−xj)

u)≥αVr(u,u), (u, P

rljrnjr⊗njr

u)≥βh(u,u).

(u, P

jljrnjr⊗njr

u)≥γh(u,u).

Sufficient condition for triangles: all the angles must be bigger than 12 degrees.

Regularity assumptions and initial data
F(t= 0,x) =−^{ε}

σ∇E(t= 0,x)

Regularity for exact solutions:V(t,x)∈W^{3,∞}(Ω) andV(t= 0,x)∈H^{3}(Ω)
Regularity for numerical initial solutions: Vh(t= 0,x)∈L^{2}(Ω)

## Uniform convergence: principle

Naive convergence estimate :||P_{h}^{ε}−P^{ε}||_{naive}≤Cε^{−b}h^{c}

Idea: Use intermediate estimates and triangular inequality (Jin-Levermore-Golse).

||P_{h}^{ε}−P^{ε}||_{L}2≤min(||P_{h}^{ε}−P^{ε}||naive,||P_{h}^{ε}−P_{h}^{0}||+||P_{h}^{0}−P^{0}||+||P^{ε}−P^{0}||)

ε→0
P_{h}^{0}

P^{ε}

h→0

P^{0}

ε→0 h→0

P^{ε}

h intermediate estimates :

||P^{ε}−P^{0}|| ≤Cε^{a},

||P_{h}^{0}−P^{0}|| ≤Ch^{d},

||P_{h}^{ε}−P_{h}^{0}|| ≤Cε^{e},
d>c,e=a.

We obtain:

||P_{h}^{ε}−P^{ε}||_{L}2≤min(ε^{−b}h^{c}, ε^{a}+Ch^{d}+Cε^{e}))
.

Comparingεandεthreshold=h^{a+b}^{ac} we obtain the final estimate:

||P_{h}^{ε}−P^{ε}||_{L}2≤h^{a+b}^{ac}
.

Physical and mathematical context
AP schemes on unstructured meshes for theP_{1}model
Nonlinear extension:M_{1}and Euler models

## Limit diffusion scheme

Limit diffusion scheme (P_{h}^{0}):

|Ω_{j}|∂tE_{j}(t)−X
r

l_{jr}(Fr,n_{jr}) = 0,
X

r

l_{jr}n_{jr}⊗n_{jr}F_{j}=X
r

l_{jr}n_{jr}⊗n_{jr}Fr,
σArFr=X

j

l_{jr}E_{j}n_{jr}, Ar=−X
j

l_{jr}n_{jr}⊗(xr−x_{j}).

ε→0

P^{0}
P^{ε}

P_{h}^{ε}

h→0
P^{0}

h

ε→0 h→0

Problemfor the estimation of

||P_{h}^{ε}−P_{h}^{0}||.

We obtain||P_{h}^{ε}−P_{h}^{0}|| ≤C^{ε}_{h}.

DA^{ε}_{h} scheme:P_{h}^{ε} with∂tFj=0.
In the estimates introduced for the
proof we replaceP_{h}^{0}byDA^{ε}_{h}.

## Limit diffusion scheme

Limit diffusion scheme (P_{h}^{0}):

|Ω_{j}|∂tE_{j}(t)−X
r

l_{jr}(Fr,n_{jr}) = 0,
X

r

l_{jr}n_{jr}⊗n_{jr}F_{j}=X
r

l_{jr}n_{jr}⊗n_{jr}Fr,
σArFr=X

j

l_{jr}E_{j}n_{jr}, Ar=−X
j

l_{jr}n_{jr}⊗(xr−x_{j}).

and

P^{0}
P^{ε}

P^{ε}

h

ε→0 h→0

P^{0}

h

∂tvh= 0

DA^{ε}_{h} ^{ε}^{→}^{0}

ε→0 h→0

Problemfor the estimation of

||P_{h}^{ε}−P_{h}^{0}||.

We obtain||P_{h}^{ε}−P_{h}^{0}|| ≤C^{ε}_{h}.
Introduction of aintermediate
diffusion scheme calledDA^{ε}_{h}.
DA^{ε}_{h} scheme:P_{h}^{ε} with∂tFj=0.

In the estimates introduced for the
proof we replaceP_{h}^{0}byDA^{ε}_{h}.

## Condition H and final result

Condition H: The discrete Hessian matrix of the solution ofP_{h}^{0}can be bounded, or
the error estimatekP_{h}^{ε}−P_{h}^{0}kcan be made independent of this discrete Hessian.

Condition H respected : we useP_{h}^{0}in the estimates.

Condition H non respected : we useDA^{ε}_{h} in the estimates

Condition H respected in Cartesian grids or on non uniform grids in 1D.

Final result: Assuming the geometrical and regularity assumptions are verified, there existC(T)>0 independent ofε, such that the following estimate holds:

kV^{ε}−V^{ε}_{h}k_{L}2([0,T]×Ω)≤Cmin
rh

ε, εmax

1, rε

h

+h+ (h+ε) +ε

!

≤Ch^{1}^{4}.
(2)
caseε≤h:kV^{ε}−V^{ε}_{h}k ≤C1min(q

ε

h,1)≤C1h
caseε≥h:kV^{ε}−V^{ε}_{h}k ≤C1min(

qh ε,

qε^{3}
h)

Introducingεthresh=h^{1}^{2} we obtainthat the worst case iskV^{ε}−V_{h}^{ε}k ≤C2h^{1}^{4}.

## Intermediary results I

V^{ε}exact solution ofP^{ε},V^{ε}_{h}numerical solution ofP_{h}^{ε}.
Estimate||V^{ε}−V^{ε}_{h}||:

We assume that the geometrical and regularity assumptions are verified. There exist a constantC>0 such that the following estimate holds:

kV_{h}^{ε}−V^{ε}k_{L}∞((0,T):L^{2}(Ω))≤C
rh

ε. Technical proof. Ideas:

Control the stability of the discrete quantitiesur andujbyε
We defineE(t) =||V^{ε}−V^{ε}_{h}||_{L}2and estimateE^{0}(t) using Young and

Cauchy-Schwartz inequalities, stability estimates, geometrical properties and a lot of calculus.

Integration in time of the estimate onE^{0}(t).

## Intermediary result II

V^{0}_{h}solution ofDA^{ε}_{h},V^{0}exact solution ofP^{0}.
Estimate||DA^{ε}_{h}−P^{0}||:

We assume that the geometrical and regularity assumptions are verified. There exist non negative constant for all timeT>0C(T) such that

||V^{0}_{h}−V^{0}||_{L}2(Ω)≤C1(T)(h+ε), 0<t≤T. (3)
Ideas of proof:

Stability estimates on the discrete quantitiesEj and the discrete gradiants at the node.

Consistency of the different discrete operators: divergence and gradient.

L^{2}estimates using the consistency errors + Gronwall Lemma.

## Intermediary result III

V^{ε}_{h}solution ofP_{h}^{ε},V_{h}numerical solution ofDA^{ε}_{h}.
Estimate||P_{h}^{ε}−DA^{ε}_{h}||:

We assume that the geometrical and regularity assumptions are verified. There exist a constant for all timeT>0,C(T) such that

||V^{ε}_{h}−Vh||_{L}2(Ω)≤C(T)εmax
1,

√
εh^{−1}

+Ch, 0<t≤T. (4)
V^{ε}solution ofP^{ε},V^{0}numerical solution ofP^{0}.

Estimate||P^{ε}−P^{0}||:

We assume that the geometrical and regularity assumptions are verified. There exist a constant for all timeT>0,C(T) such that

||V^{ε}−V^{0}||_{L}2(Ω)≤C(T)ε, 0<t≤T. (5)
Idea of Proof

WriteP^{0}=P^{ε}+R (respDA^{ε}_{h}=P_{h}^{ε}+R) withRa residue.

Obtain a upper bound of the residue byε.

L^{2}estimate of the difference between the two models or two schemes.

## AP scheme vs non AP scheme

Test case: Heat fundamental solution,ε= 0.001. Results given by hyperbolicP1

schemes on Kershaw mesh.

Diffusion solution non-AP scheme

Classical scheme Nodal scheme

## Convergence for P _{1} system

Periodic solution of theP1system dependant ofε.

E(t,x) = (α(t) +^{ε}_{σ}^{2}α^{0}(t)) cos(πx) cos(πy)

F(t,x) = −^{ε}_{σ}α(t) sin(πx) cos(πy), −^{ε}_{σ}α(t) sin(πy) cos(πx)
Mesh: random quadrangular mesh.

h/ε 1 0.01 0.001 0.00001

40-80 1.0 1.3 1.9 2.0

80-160 1.05 1.05 1.85 2.0

100-200 1.05 0.8 1.75 2.0

150-300 1.05 0.5 1.65 2.0

240-480 1.05 0.45 1.5 2.0

Hyperbolic regimeh<< ε:order 1(theoretically ^{1}_{2}).

Diffusion regimeh>> ε:order 2(theoretically 1).

Intermidiate regimeh=O(ε) :between ^{1}_{2} and ^{1}_{4}.
Future convergence analysis: ε=handε=h^{1}^{2}.

### Nonlinear extension: M

1### and Euler models

## Euler equations with friction and gravity

Work in progress.

Euler equations with gravity and friction:

∂tρ+^{1}_{ε}div(ρu) = 0,

∂tρu+^{1}_{ε}div(ρu⊗u) +^{1}_{ε}∇p= ^{1}_{ε}(ρg−^{σ}

ερu),

∂tρe+^{1}_{ε}div(ρue) + div(pu) =_{ε}^{1}(ρ(g,u)−^{σ}_{ε}ρ(u,u)).

Properties :

Entropy inequality: ∂tρS+^{1}_{ε}div(uS)≥0.

Steady states :

(E1)

u= 0,

∇p=ρg. (E2)

u= 0, ρ=ρc,

∇p=ρcg (simple case).

Diffusion limit:

∂tρ+ div(ρu) = 0,

∂tρe+ div(ρue) +pdivu= 0,
u=_{σ}^{1}

g−_{ρ}^{1}∇p
.

## AP scheme for Euler equations

Idea :

Lagrangian + remap nodal scheme coupled with the Jin-Levermore method.

Discretization:

Lagrangian part: GLACE scheme (B. Despr´es) or EUCCLYD scheme (P H. Maire).

Modification of the fluxes plugging the relation∇p=ρg−^{σ}_{ε}ρu.

Discretization of the source term using the nodal fluxes.

Nodal advection scheme for remap step.

Properties:

AP scheme with a first order limit diffusion scheme (order verified in the isothermal case with linear pressure).

Well Balanced for the simple steady state.

Positivity of the density on a CFL condition independent toε.

Entropy inequality preserved???

Well Balanced for the general steady states modifying the scheme???

## Well balanced property

Result :

If the initial data satisfy the discrete steady state∇rp=ρrgthe steady state is preserved exactly

ρr is a mean at the node of the density Continuous steady state: ρ(x),u(x) ande(x).

Question : ifρ^{0}_{j} =ρ(xj),u^{0}_{j} =u(xj) ande_{j}^{0}=e(xj) the discrete steady state is satisfied

?

forρconstant : yes.

forρvariable : probably not.

Text case with steady state as initial data: uj=0,ρj= 1 andej=_{γ−1}^{1} (xj,g).

Meshes Cartesian Random

Schemes/cells 40 80 120 40 80 120

LP-AP Ex 0 0 0 0 0 0

LP-AP SI 0 0 0 0 0 0

LP 1×10^{−8} ×10^{−9} 3×10^{−9} 0.50 0.26 0.17
L^{1}error associated to the differents schemes.

## Results for Euler equations

Test case: Sod problem withσ >0,ε= 1 andg= 0 (non longer time limit).

σ= 1

AP scheme,ρ non-AP scheme,ρ

AP scheme, non-AP scheme,

## Results for Euler equations

Test case: Sod problem withσ >0,ε= 1 andg= 0 (non longer time limit).

σ= 10^{3}

AP scheme,ρ non-AP scheme,ρ

AP scheme, non-AP scheme,

## Results for Euler equations

Test case: Sod problem withσ >0,ε= 1 andg= 0 (non longer time limit).

σ= 10^{6}

AP scheme,ρ non-AP scheme,ρ

AP scheme, non-AP scheme,

Physical and mathematical context
AP schemes on unstructured meshes for theP_{1}model
Nonlinear extension:M_{1}and Euler models

## M _{1} model and link with Euler equations

Moment model in radiative transfer : M1model.

∂tE+^{1}_{ε}divF= 0,

∂tF+_{ε}^{1}∇(bP) =−^{σ}

ε^{2}F,

∂tF+^{1}_{ε}div(u⊗F) +_{ε}^{1}∇q=−^{σ}

ε^{2}F.

(6)

E energy,Fthe flux andPb=^{1}_{2}((1−χ(f))Id+ (3χ(f)−1)^{f⊗f}_{kfk})E the pressure.

f=||F||/Eandχ(f) = ^{3+4f}^{2}

5+2√

4−3f^{2}.
Properties :

Diffusion limit,ε→0 :∂tE−div(_{3σ}^{1}∇E) = 0,
Entropy inequality: ∂tS+^{1}_{ε}div(Q)≥0,
Maximum principle: E>0,||f||<1.

Formulation like dynamic gas system:

F=uE+qu, Pˆ=u⊗F+qbId, q= ^{1−χ}_{2} E,u=^{3χ−1}_{2} _{||f||}^{f}_{2}.

## M _{1} model and link with Euler equations

Moment model in radiative transfer : M1model.

∂tE+^{1}_{ε}divF= 0,

∂tF+_{ε}^{1}∇(bP) =−^{σ}

ε^{2}F,

⇐⇒

∂tE+_{ε}^{1}div(Eu+qu) = 0,

∂tF+^{1}_{ε}div(u⊗F) +^{1}_{ε}∇q=−^{σ}

ε^{2}F.

(6)

E energy,Fthe flux andPb=^{1}_{2}((1−χ(f))Id+ (3χ(f)−1)^{f⊗f}_{kfk})E the pressure.

f=||F||/Eandχ(f) = ^{3+4f}^{2}

5+2√

4−3f^{2}.
Properties :

Diffusion limit,ε→0 :∂tE−div(_{3σ}^{1}∇E) = 0,
Entropy inequality: ∂tS+^{1}_{ε}div(Q)≥0,
Maximum principle: E>0,||f||<1.

Formulation like dynamic gas system:

F=uE+qu, Pˆ=u⊗F+qbId, q= ^{1−χ}_{2} E,u=^{3χ−1}_{2} _{||f||}^{f}_{2}.

## Scheme and properties

Idea :

Use the proximity between the new formulation (previous slide) and the Euler equations to use a Lagrange+remap AP scheme.

Link with the Euler equations:

Introduction of artificial densityρ

Introduction of Thermodynamics quantities:E=ρe,F=ρvandS=ρs.

We apply the AP scheme for hydrodynamics equations obtained and after we obtain a scheme for the quantitiesE andF.

Properties:

AP schemewith a first order nonlinear limit diffusion scheme.

Numerical stability independent toε(verified numerically).

Second order limit diffusion schemeusing a MUSCL procedure in the remap step.

Entropy inequality preservedby the semi-discrete scheme.

Maximum principle preservedby the scheme without MUSCL in the transport regime (ε=O(1)).

## Results for M _{1} model

Diffusion test case: The data areE(0,x) =G(x) withG(x) a Gaussian andσ= 1.

Final timeTf = 0.011.

Schemes NL VF5 Linear M1

Meshes order E_{j}>0 order E_{j}>0 order E_{j}>0 order E_{j}>0

Cartesian 1.9 yes 2 yes 2 yes 2.0 yes

Rand. quad 1.9 yes 0.3 yes 1.98 no 2. yes

Regular. tri. 2.2 yes 2 yes 2. yes 2.0 yes

Rand. tri. 2.15 yes 1. yes 1.32 no 1.9 yes

Kershaw 1.9 yes 0 yes 2 no 1.9 yes

NL : Limit diffusion scheme ofM1scheme. M1: AP scheme forM1model with
ε= 10^{−3}.

Discrete Maximum principle: the data areσ= 0,E(0,x) =Fx(0,x) =1_{[0.4:0.6]}2 et
Fy(0,x) = 0. The solution isE(t,x) =Fx(t,x) =1[0.4+t:0.6+t]^{2} andFy(t,x) = 0.

Meshes order Ej>0 kfjk<1

Cartesian 0.5 yes yes

Rand. quad 0.5 yes yes

Kershaw 0.49 yes yes

## Conclusion and future works

Conclusion

P1model: AP nodal scheme on distorted meshes with a stability independant ofε.

P1model: Uniform convergence for the semi discrete scheme on unstructured meshes.

Non linear model: AP scheme with maximum principle for theM1and extension for the Euler equations.

All models: Spurious mods in few cases (example: Cartesian mesh + initial Dirac data).

Future works

Numerical convergence analysis for test cases with nonlinear diffusion limit (full Euler and isothermal Euler equations).

Analysis of theP1AP discretization: Time convergence, CFL condition.

Analysis of the Euler AP discretization: entropy stability.

Extension to Euler scheme for non constant gravity and more complicated steady states.

Generic stabilization procedure the for nodal schemes.

Others works

AP schemes for generic linear systems with source terms (PN,SN models) using

”micro-macro” decomposition.

AP scheme forP1model based on the MPFA diffusion scheme.

## Thank you

Thank you for your attention