Analysis of Asymptotic Preserving schemes with the modified equation - - PowerPoint PPT Presentation

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno Despr´ es (LJLL-UPMC) collaboration with

• C. Buet (CEA) et E. Franck (PhD-CEA-LJLL)

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

• p. 1 / 25

Model problem Modified equations Main result Schemes Numerical results

1D telegraph equation

Model problem (stiff source terms, radiation , friction, . . .)          ∂tuε + 1 ε∂xvε = 0, ∂tvε + 1 ε∂xuε + σ ε2 vε = 0. σ > 0 is a given coefficient. The small parameter is 0 < ε ≤ 1. Hilbert expansion with respect to the small parameter ε uε = u0 + · · · , vε = v0 + εv1 + · · · yields v0 = 0 and ∂xu0 + σv1 = 0. It justifies the limit diffusion equation ∂tu0 − 1 σ∂xxu0 = 0. Jin-Levermore, Numerical methods for hyperbolic conservation laws with stiff relaxation terms, JCP 1996.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Standard discretization

The usual F.V scheme (Riemann) writes        uj − uj ∆t + vj+1 − vj−1 2ε∆x − uj+1 − 2uj + uj−1 2ε∆x = 0, vj − vj ∆t + uj+1 − uj−1 2ε∆x − vj+1 − 2vj + vj−1 2ε∆x + σ ε2 uj = 0. Hilbert expansion (uj = u0

j + · · · and vj = v0 j + εv1 j + · · · )

yields v0

j = 0,

u0

j+1 − u0 j−1

2∆x + σv1

j = 0.

The limit scheme is uj 0 − u0

j

∆t − 1 σ u0

j+2 − 2u0 j + u0 j−2

∆x2 − ∆x 2ε u0

j+1 − 2u0 j + u0 j−1

∆x2 = 0.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Summary

• The limit scheme is consistent with

∂tu − 1 σ + ∆x 2ε

• ∂xxu = 0.

This is not correct in the regime ∆x ≥ ε ! !

• The main objective of A.P. methods is to modify the basic

scheme.

• There exists different approaches : Jin-Levermore,

Gosse-Toscani, . . ., Jin-Pareschi, . . .. The structure of these methods may seem strange, in particular for the most efficient methods.

• First goal of this presentation : can we understand a priori

the necessary modifications, starting from PDEs ?

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Modified equations

• Start from the modified equation

     ∂tuε,α + 1 ε (∂xvε,α − α∂xxuε,α) = 0, ∂tvε,α + 1 ε (∂xuε,α − α∂xxuε,α) = − σ ε2 vε,α, with two small parameters : ε and α ≈ ∆x

2 .

• Reminder : Hilbert expansion with respect to ε (α > 0 fixed)

is not correct ∂tu − 1 σ + α ε

• ∂xxu = 0

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Modified equation of the first kind

• Introduce the Magic coefficient M =

ε ε+σα ∈]0, 1] together

with      ∂tuε,α + M ε (∂xvε,α − α∂xxuε,α) = 0, ∂tvε,α + 1 ε (∂xuε,α − α∂xxvε,α) = − σ ε2 vε,α,

• Hilbert expansion with respect to ε (α > 0 fixed) yields

∂tu0,α − M 1 σ + α ε

• ∂xxu0,α = 0.

This is correct now since : M 1

σ + α ε

• =

ε ε+ασ ε+σα σε

= 1

σ.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Modified equation of the second kind

• The symetry in the equations is better using the Magic

coefficient in both equations      ∂t uα,ε + M ε (∂x vα,ε − α∂xx uα,ε) = 0, ∂t vα,ε + M ε (∂x uα,ε − α∂xx vα,ε) = −σM ε2 vα,ε,

• Once again, Hilbert expansion is correct

∂t uα,0 − 1 σ∂xx uα,0 = 0.

• The identity energy is correct in the sense
• R
• uα,ε(t)2 +

vα,ε(t)2 2 dx ≤

• R
• uα,ε(0)2 +

vα,ε(0)2 2 dx.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Main result

An extension of a result proved for numerical methods in Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes,

• Numer. Math., 2012, with C. Buet and E. Franck yields

Theor : Consider well prepared data u0 ∈ H3(R), v0 = − ε σ∂xu0 + ε2v2 v2 ∈ H1(R). There exists C > 0 ind´ ependent of ε such that (t ≤ T) uε(t) − uα,ε(t)L2(R) + vε(t) − vα,ε(t)L2(R) ≤ C min(α, ε).

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Sketch of the proof

Define e = uε − uα,ε and f = vε − vα,ε One has      ∂te + M ε (∂xf − α∂xxe) = r, ∂tf + M ε (∂xf − α∂xxf ) + σM ε2 f = s, with r = ∂tuε + M ε (∂xvε − α∂xxuε) = (1 − M)∂tuε − Mα ε ∂xxuε, s = ∂tvε + M ε (∂xuε − α∂xxvε) + σM ε2 vε = (1 − M)∂tvε − Mα

ε ∂xxvε.

Main intermediate result rL∞([0,T]:L2(R)) + sL2([0,T]:L2(R)) ≤ C min(α, ε) which proves the claim.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Philosophy

An A.P. scheme with good properties introduces the Magic coefficient M at the right place. The design principle of the A.P. scheme is less important. It may be a full Riemann solver, a simplified one, coming from W.B. (well balanced) techniques, a direct finite difference scheme, . . .

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Jin-Levermore

Riemann invariants v ± u are modified vn

j + un j

= vn

j+ 1

2 + un

j+ 1

2

+ σ∆x

2ε vn j+ 1

2 ,

vn

j+1 − vn j+1

= vn

j+ 1

2 − un

j+ 1

2

+ σ∆x

2ε vn j+ 1

2 .

The solution of this linear system is    vn

j+ 1

2 = M

2

• vn

j + vn j+1 + un j − un j+1

• ,

un

j+ 1

2 = 1

2

• un

j + un j+1 + vn j − vn j+1

• .

where M =

ε ε+ ∆x

2 σ is the correct value. Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Final scheme

Since vn

j+ 1

2 = MvClassique

j+ 1

2

, one obtains the Jin-Levermore scheme      uj − uj ∆t + M vj+1 − vj−1 2ε∆x −M uj+1 − 2uj + uj−1 2ε∆x = 0, vj − vj ∆t + uj+1 − uj−1 2ε∆x −vj+1 − 2vj + vj−1 2ε∆x + σ ε2 vj = 0. It corresponds to the modified equation of the first kind.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

• p. 12 / 25

Model problem Modified equations Main result Schemes Numerical results

Gosse-Toscani scheme

Moreover replace the source term σ

ε2 vj by

σ ε2 vj− 1

2 + vj+ 1 2

2 . One obtains exactly      uj − uj ∆t + M vj+1 − vj−1 2ε∆x − M uj+1 − 2uj + uj−1 2ε∆x = 0, vj − vj ∆t + M uj+1 − uj−1 2ε∆x − M vj+1 − 2vj + vj−1 2ε∆x + M σ ε2 vj = 0, It corresponds to the modified equation of the second kind.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

• p. 13 / 25

Model problem Modified equations Main result Schemes Numerical results

MultiD : uε ∈ R, vε ∈ R2

Solve : ∂tuε + 1 ε∇ · vε = 0, ∂tvε + 1 ε∇uε = − σ ε2 vε.

0.5 1 1.5 2 0.5 1 1.5 2 0.5 1 1.5 2 0.5 1 1.5 2

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Jin-Levermore on edges

xj xr+1 xr−1 l jk xr Cell Ω j Cell Ωk xk njk

First problem : xjxk is not parallel to normal vector njk.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Jin-Levermore on edges

The natural F.V. generalisation of the 1D A.P. scheme is |Ωj| ∂tuj(t) + 1

ε

• k ljk(vjk, njk) = 0,

|Ωj| ∂tvj(t) + 1

ε

• k ljkujknjk = − |Ωj| σ

ε2 vj,

with Jin-Levermore’s procedure in the normal direction njk ujk − uj + (vjk − vj, njk) = − σ

ε djk(vjk, njk),

ujk − uk − (vjk − vk, njk) = σ

ε dkj(vjk, njk).

Here djk = dist (xj, xjk) is the distance to the edge.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Formal limit

The limit ε → 0 is the FV5 scheme |Ωj| ∂tuj(t)−

• k

ljk 1 σ uk − uj djk + dkj = 0, with djk+dkj = d(xj, xk). This scheme is correct for Delaunay meshes. If xjxk is parallel to normal vector njk : OK.

xj xr+1 xr−1 l jk xr Cell Ω j Cell Ωk xk njk

This mesh is not Delaunay. Edge based Jin-Lervermore not A.P. on general meshes.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Nodal scheme : r is the index of a vertex

xj xr+1 xr−1 xr Cell Ω j Cell Ωk l jrnjr

Definition : ljrnjr = (xr−1xr+1)⊥

2

; njr is the corner normal, ljr is the length of the corner. More general definition : ljrnjr = ∇xr Sj. Corner or point fluxes receive increasing interest (Roe). Corner scheme : |Ωj| ∂tuj(t) + 1

ε

• r ljr(vr, njr) = 0

|Ωj| ∂tvj(t) + 1

ε

• r ljrujrnjr = − |Ωj| σ

ε2 vj

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Vertex-based solvers

The node xr is given. The unknowns of the linear system are (ujr)j and vr.

• Lin. system :

ujr = uj + (vj − vr, njr),

• j ljrujrnjr = 0.

xj x 1

xr xj− 1

Vr

This linear system is non singular on many meshes.  

j

ljr(njr ⊗ njr   vr =

• j

ljr (ujnjr + njr ⊗ njrvj)

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Jin-Levermore and vertex-based solvers

• ujr = uj + (vj − vr, njr) − σ

ε (vr, xr − xj),

• j ljrujrnjr = 0.

Good news : The linear system is still non singular on ”almost” all meshes  

j∈cells

ljr

• njr ⊗
• njr + σ

ε (xr − xj)

• 



• =Ar

vr =

• j

ljr (ujnjr + njr ⊗ Bad news : Algebra shows that |Ωj|u′

j(t) − r ljr

• njr, Mrvclassical

r

• = 0 where Mr is a matrix

Mr = A−1

r

 

j∈cells

ljrljrnjr ⊗ njr   .

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Limit scheme for ε → 0+

It writes        u′

j(t) −

1 |Ωj|

• r

ljr (njr, vr) = 0, Arvr =

• j

ljrnjruj, with Ar = −

• j ljrnjr ⊗ (xr − xj)
• .
• Theor. (very good news) : Under reasonnable technical

conditions, the limit diffusion scheme is convergent for all time T > 0. There exists a constant C(T) > 0 such that e(t)L2(Ω) + fL2([0,t]×Ω) ≤ C(T)h, 0 < t ≤ T. In this sense the Jin-Levermore nodal based scheme is A.P. The modified equation is much more complicated to undertsand than the 1D case.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

2D numerical example

Fundamental solution computed on the Kershaw mesh : ε = 10−3.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

A radiation problem

Here the opacity σ is large in the squares and small everywhere else. The methods studies in this presentation work perfectly for non constant coeeficients.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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Model problem Modified equations Main result Schemes Numerical results

Conclusion

Modified equation techniques in 1D show a property of uniform approximation for A.P. methods. Adaptation in dimension d > 1 reveal new and difficult problems. To solve these problems, strong modifications of the usual Finite Volume setting are needed. Nodal Finite Solver are attractive in multiD. See : Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes Numerische Math., mars 2012, with C. Buet and

• E. Franck

Many open problems left.

Analysis of Asymptotic Preserving schemes with the modified equation 2D extension on general meshes Bruno

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