Systematic testing of explicit positivity preserving algorithms for the heat-equation

Abstract

In this work, we performed systematic tests of recently invented stable and explicit algorithms which preserve the positivity of the solution for the linear heat equation. It is well known that the widely used explicit finite difference schemes are typically unstable if the time step size is below the so called CFL limit, and even if they are stable, they can produce negative temperatures. However, the numerical solutions should satisfy the same properties as the exact solution, such as positivity. Thus, we collected the available explicit positivity preserving methods, most of them created by us recently to examine their performance and relative competitiveness. We tested them in the case of several 2D systems to find how the errors depend on the stiffness ratio and the CFL limit of the system for each algorithm. Then we created an anisotropic but equidistant grid by shrinking the vertical dimension of the 2D system and examined how this kind of anisotropy effects the errors.