New stable, explicit, second order hopscotch methods for diffusion-type problems

The aim of this paper is to systematically construct and test novel odd-even hopscotch-type numerical algorithms solving the diffusion or heat equation. Among the studied explicit two-stage methods some of them are unconditionally stable and have second order convergence rate in time step size, which is proved analytically as well. We apply the best methods to the nonlinear Fisher's equation to demonstrate that they work also for nonlinear equations. Then, in order to examine the competitiveness of the new algorithms, we test them for the heat equation against widely used numerical solvers in cases where the media are strongly inhomogeneous and thus the coefficients strongly depend on space. The results suggest that the new methods are significantly more effective than the widely used explicit or implicit methods, especially for extremely large stiff systems.(c) 2023 The Author(s). Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in Simulation (IMACS). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).