New explicit algorithm based on the asymmetric hopscotch structure to solve the heat conduction equation

Abstract

In this paper we will consider a new four-stage structure inspired by the well-known odd-even hopscotch method to construct new schemes for the numerical solution of the two-dimensional heat or diffusion equation. In this structure the first and the last time step are halved stage and therefore the time steps are shifted compared to each other for odd and even cells. We insert 10 concrete formulas into this structure to obtain 10⁴ different combinations. First we test all of these in case of small systems with random parameters, and then examine the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. We select the top 5 combinations, and demonstrate that these new methods are indeed effective if the goal is to produce results with acceptable accuracy in very short time.