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

TL;DR

This paper develops explicit third and fourth order accurate Multi-Step Runge-Kutta (MSRK) methods and outlines 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.

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.

Figures (8)

• Figure 1: Absolute stability regions (ASRs) for RK4-2(1), RK4-2(2) and RK4-3 schemes given by the coefficients in Table \ref{['tab:grand_coefficient_table']}. For comparison, we also plot the ASR of the standard RK4 method. The $x$ and $y$ axes represent, respectively, the real and imaginary parts of $z$, denoted by $\Re(z)$ and $\Im(z)$.
• Figure 2: Left: Convergence tests for the evolution of a scalar wave equation using our new methods. Right: Evolution of the $\ell_\infty$ norm of the error in $\Pi$ versus the number of time steps taken (which was $36800$ or $100$ oscillations), computed with RK4 and our methods. The amplitude of the wave is $A=1$.
• Figure 3: Example RST results for a passing CFL of $0.46$ using RK4-2(1).
• Figure 4: Log-log plots of the time evolution of the $\ell^2$ norm of the Hamiltonian constraint (left) and a log-log plot of the relative differences of that quantity obtained with our methods to that obtained with RK4 (right)
• Figure 5: $r \Psi_4(2,2)$ at $r=90$ evolution obtained with our methods and RK4 (left) and relative differences differences of the same signal obtained with our methods with that obtained with R4K on a log scale in the $y$ axis