A nonlinear convex combination in the construction of finite volume scheme satisfying maximum principle

Abstract

In the construction of cell-centered finite volume schemes for general diffusion equations with discontinuous coefficients on distorted meshes, auxiliary unknowns are usually introduced to resolve discontinuities across cell-edges. In general, it is difficult to express unconditionally auxiliary unknowns as a convex combination of primary unknowns around. As we know, all existing methods have to impose certain restrictive conditions on cell-distortion and coefficient discontinuities, especially when designing discrete schemes satisfying the maximum principle. In this paper, we present a nonlinear convex combination method, in which each vertex unknown is eliminated by a nonlinear convex combination of two cell-centered unknowns, which are respectively the maximum and minimum of those cell-centered unknowns around the vertex. As two application examples we present two cell-centered schemes satisfying the maximum principle based on the nonlinear convex combination. An analysis for the discrete flux shows that our scheme can exactly recover the linear solution. Numerical results are presented to show the accuracy of the resulting schemes and verify the discrete maximum principle.