Stable, explicit numerical methods for solving heat conduction and diffusion problems

Abstract

In this paper, we present and compare mainly recently published explicit and stable numerical methods with each other and with an implicit method for solving the linear heat conduction or diffusion equation. Unlike standard explicit methods (such as FTCS or Runge-Kutta), these methods use a non-polynomial expression of the time step size to calculate the new values of the variable. The performance of the methods is compared via such non-trivial analytical solutions where the diffusion coefficient depends on the spatial variable. In the cases studied, the best new methods perform much better than the traditional Runge-Kutta methods.