Building monotone corrections for cell-centered finite volume schemes 1 2 3 Cl´ement Canc`es , Mathieu Cathala , Christophe Le Potier

1

Universit´e Pierre et Marie Curie, 2Universit´e Montpellier 2, 3CEA Saclay

Statement of the problem

How to reach monotonicity ? 1. Estimate the bad contributions of the initial scheme :

Approximate the solution u ¯ of  − div(D∇¯ u) = f in Ω, u ¯ = 0 on ∂Ω.

I

Rewrite the initial operator as X AK(u) = ? |uK − uL| AK(u)

(? =positive coef.)

L∈V(K)

No discrete maximum principle for the classical linear discretization schemes I Aim : Designing corrections for a cell-centered finite volume scheme which I

2. Compensate these contributions : Choose βK,L such that βK,L(u) ≥ ? |AK(u)|. I Corrected equations : X
βK,L(u) + ?sgn(uL − uK)AK(u) (uK − uL)

I

provide a Discrete Maximum Principle I retain the main properties of the scheme I

L∈V(K)

The initial scheme I

How to maintain conservativity and coercivity ?

M mesh of Ω, mesh size : h.

I

Discrete unknowns : I

L

K

I

uK ≈ u ¯|K

~nK,σ · xK

Conservativity is maintained under the symmetry condition βK,L = βL,K I Coercivity is maintained under the positivity condition βK,L ≥ 0

Approximate solution u = (uK) :

I

Approximation of the fluxes on the edges : Z D∇¯ u · ~nK,σ , FK,σ + FL,σ = 0 FK,σ ≈

Examples of corrections ([2], [3])

σ

σ I

Discrete operator A(u) : Z X AK(u) = FK,σ ≈ div(D∇¯ u) K

σ∈EK

Initial scheme : Z ∀K,

−AK(u) =

f K

Different choices for ? and different ways to achieve symmetry : |AK(u)| |AL(u)| βK,L(u) = P +P J∈V(K) |uK − uJ| J∈V(L) |uL − uJ|   |AK(u)| |AL(u)| βK,L(u) = max , CardV(K) |uK − uL| CardV(J) |uK − uL| I A discrete maximum principle is recovered : I

(2) (3)

Main properties of the initial scheme I

Conservativity : each equation writes as a balance of numerical fluxes X AK(u) = FK,σ σ∈EK

I

Coercivity of the discrete operator −A : I

For any discrete function u = (uK)K, X − AK(u)uK ≥ C kuk2D

correction (2) (min. value 1.82 × 10−15)

(k·kD : discrete H10 norm)

What about convergence as h → 0 ?

K∈M I

Consistency with the continuous operator I

As h → 0, if (uh) is a sequence of discrete functions such that (kuh kD ) is bounded and uh → u ¯ in L2 then : Z X ∀ϕ ∈ Cc∞(Ω), − AK(uh)ϕ(xK) −−→ D∇¯ u · ∇ϕ K∈M

h→0

Ω

Lack of maximum principle I

We consider a sequence of meshes such that h → 0. (uh) sequence of corresponding numerical solutions of the corrected schemes. h I Coercivity confers compactness : u → u ¯ Question : does u ¯ solve the continuous problem ? ϕ : test function. R h h I Passage to the limit in the corrected scheme −AK(u ) + RK(u ) = Kf : Z X X X RK(uh)ϕ(xK) = fϕ(xK) • − AK(uh)ϕ(xK) +

2

K∈M

Illustration on Ω =]0, 0.5[ with the initial scheme ( from [1] :   −6 2

2

−6

10 x + y (10 − 1)xy I D(x, y) = and f = −6 2 −6 2 (10 − 1)xy x + 10 y I Grid of made of 4096 squares (h = 1/128). 1 x2+y2

correction (3) (min. value 7.95 × 10−10)

10 in ]0, 25, 0.5[2 0 elsewhere

h→0

K∈M consistency

?

Z •

Z D∇¯ u · ∇ϕ

+

0

=

Ω I

Ω

Conservativity of the correction : X X X RK(uh)ϕ(xK) ≤ C diam(K) βK,L(uh) uhK − uhL K∈M

I

minimum value −4.71 × 10

position of the undershoots

K∈M

Sufficient condition : X

diam(K)

K∈M

Monotonicity and discrete maximum principle I

h h βK,L(u ) uK − uL −−→ 0 h

h→0

L∈V(K)

X

(4)

h diam(K) AK(u ) −−→ 0

K∈M I

h→0

Correction (3) : Check there is at least one “good” contribution of the initial part ∃J ∈ V(K),

Corrected schemes

AK(uh)(uhJ − uhK) > 0

References

I

Workshop on Complex grids and fluid flows, April 2-4th, 2012

X

L∈V(K)

⇒ Convergence of the corrected scheme under the condition (4). I Convergence conditions for the two examples : how to satisfy (4) ? I Correction (2) : look at the behavior of the initial part

Monotonicity : each equation can be written Z X τK,L(u) (uK − uL) = f K L∈V(K) >0 ⇒ Discrete maximum principle

Idea : Correct the equations in a monotone way : Z X f −AK(u) + βK,L(u)(uK − uL) = K L∈V(K) corrective terms initial scheme Question : How to build the coefficients βK,L(u) such that I Monotonicity is achieved ? I The main properties of the initial scheme A still remain ?

fϕ

Behavior of the corrective term as h → 0 ? I

−3

K

K∈M

(1)

I. Aavatsmark, T. Barkve, O. Boe, T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods, Siam J. Sci. Comput., 19 (1998), no. 5, 1700–1716. C. Canc`es, M. Cathala, C. Le Potier, Monotone coercive cell-centered finite volume schemes for anisotropic diffusion equations, HAL: hal-00643838, 2011. C. Le Potier. Correction non lin´eaire et principe du maximum pour la discr´etisation d’op´erateurs de diffusion avec des sch´emas volumes finis centr´es sur les mailles., C. R. Acad. Sci. Paris, 348 (2010), no. 11-12, 691–695.

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