Discrete Maximum Principle and Adequate Discretizations of Linear Parabolic Problems

In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet-type boundary condition. First we formulate the qualitative properties for the differential equations and shad light on their relations. Then we show how the well-known discretization schemes can be written in the form of a one-step iterative process. We give necessary and sufficient conditions of the main qualitative properties of these iterations. We apply the results to the finite difference and Galerkin finite element solutions of linear parabolic problems. In our main result we show that the nonnegativity preservation property is equivalent to the maximum-minimum principle and they imply the maximum norm contractivity. In one, two and three dimensions, we list sufficient a priori conditions that ensure the required qualitative properties. Finally, we demonstrate the above results on numerical examples.