New third order low-storage SSP explicit Runge-Kutta methods

Abstract

When a high dimension system of ordinary differential equations is solved numerically, the computer memory capacity may be exhausted. Thus, for such systems, it is important to incorporate low memory usage to some other properties of the scheme. In the context of strong stability preserving (SSP) schemes, some low-storage methods have been considered in the literature. In this paper we study 5-stage third order 2N low-storage SSP explicit Runge-Kutta schemes. These are SSP schemes that can be implemented with 2N memory registers, where N is the dimension of the problem, and retain the previous time step approximation. This last property is crucial for a variable step size implementation of the scheme. In this paper, first we show that the optimal SSP methods cannot be implemented with 2N memory registers. Next, two non-optimal SSP 2N low-storage methods are constructed; although their SSP coefficients are not optimal, they achieve some other interesting properties. Finally, we show some numerical experiments.