Accelerating Numerical Relativity Simulations with New Multistep Fourth-Order Runge-Kutta Methods

TL;DR

This paper introduces explicit multistep fourth-order Runge-Kutta methods that reuse previous data to reduce computational costs, validated in Numerical Relativity simulations, with potential applications in other differential equation solvers.

Contribution

The paper develops and tunes new multistep RK4 methods that require fewer evaluations, improving efficiency in large-scale differential equation simulations.

Findings

Fewer intermediate stage evaluations needed per step.

Enhanced stability regions allow larger time steps.

Validated improvements in Numerical Relativity applications.

Abstract

Many HPC applications that solve differential equations rely on the Runge-Kutta family of methods for time integration. Among these methods, the fourth-order accurate RK4 scheme is especially popular. This time integration scheme requires applications to evaluate four intermediate stages to take one time step. Depending on the complexity of the problem being solved, the evaluation of these intermediate stages can be computationally expensive. In this paper we develop explicit fourth-order accurate Multistep Runge-Kutta (MSRK) methods. The advantage of such methods is that they re-use data from previous time steps, thus requiring fewer intermediate stage evaluations and potentially speeding up applications. We outline a procedure to obtain and tune the method's coefficients by adjusting their stability regions in an attempt to maximize the size that a time step can take. We validate and evaluate our new methods in the context of Numerical Relativity applications using the EinsteinToolkit. We believe, however, that these methods and results should generalize to other applications using explicit Runge-Kutta methods.