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

TL;DR

This paper develops and analyzes low-memory, third-order explicit Runge-Kutta methods with strong stability preservation, focusing on 2N* storage schemes suitable for high-dimensional ODE systems, including construction and numerical testing.

Contribution

It introduces two new 2N* low-storage SSP explicit Runge-Kutta methods and analyzes their properties, addressing the challenge of memory constraints in high-dimensional ODEs.

Findings

Optimal SSP methods cannot be implemented with 2N* memory.

Constructed two non-optimal 2N* SSP methods with interesting properties.

Numerical experiments demonstrate the methods' performance.

Abstract

When a high dimension system of ordinary differential equations is solved numerically, the computer memory capacity may be compromised. 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.