‘Discrete Maximum Principle Honored by Conventional Finite Volume Schemes for Diffusion-Convection-Reaction Problems: Proof with Geometrical Arguments

Abstract

From the Engineering point of view, the Maximum Principle is physically an important property met by solutions of elliptic partial differential equations (PDE for short) of second order governing diffusion-convectionreaction phenomena. This property is also called Positivity-Preserving Property in the literature. At the discrete level the Positivity-Preserving Property is required for any numerical scheme designed for solving such PDE. By means of algebraic arguments it is well-known that conventional finite volume schemes for second order elliptic PDE meet the discrete maximum principle. In this communication we expose a new technique based upon geometric arguments for proving that conventional finite volume schemes for diffusion-convection-reaction problems meet the discrete version of Maximum Principle. Notice that the above mentioned geometrical technique works for any space dimension.