A Second-Order Cartesian Grid Finite Volume Technique for Elliptic Interface Problems
A second-order finite volume method (FVM) difference scheme for elliptic interface problems is discussed. The method uses bilinear functions on Cartesian grid for the solution resulting in a compact nine-point stencil. Numerical experiments show second or
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Abstract. A second-order finite volume method (FVM) difference scheme for elliptic interface problems is discussed. The method uses bilinear functions on Cartesian grid for the solution resulting in a compact nine-point stencil. Numerical experiments show second order of accuracy. 2000 Mathematics Subject Classification: 65M06, 65M12. Keywords: parabolic equations, dynamical boundary conditions.
1
Introduction
In this paper we construct a finite-difference scheme that has second order of accuracy for an elliptic equation of the form ∇(β(x, y)∇u) − k(x, y)u = f (x, y), (x, y) ∈ Ω\Γ
(1)
with an embedded interface Γ . For simplicity we assume Ω to be a rectangle and impose Dirichlet boundary conditions. The curve Γ separates two disjoint + − + − sub-domains Ω and Ω with Ω = (Ω ∪ Ω )\Γ , see Figure 1 (a) for an illustration. Along the interface Γ we prescribe jump conditions of a generalized proper lumped source: + + r− (x, y)u− n + r (x, y)un = [u] + g1 (x, y), (x, y) ∈ Γ
(2)
[β(x, y)un ]Γ = δ(α+ (x, y)u+ − α− (x, y)u− ) + g2 (x, y), (x, y) ∈ Γ,
(3)
where the symbol [v] stands for the jump of the function v across Γ , i.e., [v] = v + − v − , v + (x, y) =
lim
ζ→(x,y),ζ∈Ω +
v(ζ), v − (x, y) =
lim
ζ→(x,y),ζ∈Ω −
v(ζ), (4)
r− (x, y), r+ (x, y) and δ are given nonnegative functions. Moreover, r− and r+ do not simultaneously vanish, α+ = r− /(r− + r+ ), α− = r+ /(r− + r+ ), un = (∇u.n) and n = (n 1 , n 2 ) is the unit normal vector on Γ , pointing from Ω + to Ω − . Note that, for g1 = 0 and δ = 0, the transmission conditions (2) and (3) become the nonhomogeneous conditions of nonperfect contact [5,6]. I. Lirkov, S. Margenov, and J. Wa´ sniewski (Eds.): LSSC 2007, LNCS 4818, pp. 679–687, 2008. c Springer-Verlag Berlin Heidelberg 2008
680
J.D. Kandilarov, M.N. Koleva, and L.G. Vulkov
In the literature one can find a great number of different approaches for the numerical solution of elliptic interface problems. We limit our discussion here to the immersed interface method (IIM). It is a second order finite difference method on Cartesian grids for second order elliptic and parabolic equations with variable coefficients, see [2]. The finite element IIM, based on Cartesian triangulations, is developed in [2,3]. Starting with an idea of P. Vabischevich [6], for piecewise linear approximation of the interface curve in the integrointerpolation method (≡ FVM), we use as in [4] piecewise bilinear functions on Cartesian grid, which makes our method similar to the finite element method FEM. To obtain finite volume formulation of (1)-(4), we integrate the equation (1) ¯ and for irregular control volumes (interover an arbitrary control volume e ∈ Ω sected by the interface) we have β∇u · ndS − kudV = f dV + f dV − [βun ]dS, (5) ∂e
e
e+
e−
Γe
whereΓe is the part of the embedded interface Γ , lying inside e and ∂e = (∂e+ ∂e− ) \ Γe . The finite volume method (FVM) is based on a ’balance’ approach and originates from the integrointerpolation difference scheme method of Samarskii [5], designed first of all to be locally con
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