Some properties of the Runge-Kutta-Legendre super-time-stepping explicit methods

Abstract

In this paper we will show that the Runge-Kutta-Legendre (RKL) super-time-step methods are built up in stages by combining forward Euler steps with linear extrapolation steps. For second order, we will show that linear interpolation is also used. By using these characteristics a simplified algorithm will be presented. The effect of different types of external boundary conditions are shown. For Neumann (zero-flux) and Periodic the methods are shown to be monotone. For Dirichlet it is shown that there are regions of non-monotonicity where solutions have the potential to go negative. These solutions are nonphysical and will lead to erroneous results if they are feed back into system. To remove these limitations two solution strategies are presented based on different non-uniform fixed meshing philosophies. A number of applications are shown with solutions validated against analytic. For a monotone heat front and a diffused heat pulse, the RKL results are shown to be physically correct and computationally cheaper. For a compact heat pulse it will be shown that adverse effects can occur if the number of steps is too large. It will be shown that results are significantly improved by reducing the number of steps and increasing the number of outer cycles. For general applications the universal approach is to try different numbers of steps and then study any sensitivities. © British Crown Copyright AWE/2020