Existence of solution of a finite volume scheme preserving maximum principle for diffusion equations

Abstract

In this article, a cell‐centered finite volume scheme preserving maximum principle for diffusion equations with scalar coefficients is developed. The construction of the scheme consists of three steps: at first the discrete normal flux is obtained by a linear combination of two single‐sided fluxes, then the tangential term of the normal flux is modified by using a nonlinear combination of two single‐sided tangential fluxes, finally the auxiliary unknowns in the tangential fluxes are calculated by the convex combinations of the cell‐centered unknowns. It is proved that this nonlinear scheme satisfies the discrete maximum principle (DMP). Moreover, the existence of a solution of the nonlinear scheme is proved by using the Brouwer's fixed point theorem and the bounded estimates. Numerical experiments are presented to show that the scheme not only satisfies DMP, but also obtains the second‐order accuracy and conservation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 80–96, 2018