Testing and improving a non-conventional unconditionally positive finite difference method

Abstract

In this article, we applied the Unconditionally Positive Finite Difference (UPFD) method of Chen-Charpentier and Kojouharov to a diffusion-type partial differential equation (PDE), the so-called heat equation. Stability and convergence properties have been verified numerically for specific initial and boundary conditions. We also elaborated and tested the UPFD method with adaptive step size control. We have demonstrated that this version can be more efficient by reducing the time needed for solving the heat equation in one dimension.