Mixed-precision explicit stabilized Runge-Kutta methods for single- and multi-scale differential equations

TL;DR

This paper introduces mixed-precision Runge-Kutta-Chebyshev methods that combine low- and high-precision computations to efficiently solve differential equations without losing accuracy or stability.

Contribution

The authors develop a new class of mixed-precision RKC schemes that maintain convergence order and stability while reducing computational cost, especially for multiscale problems.

Findings

Mixed-precision RKC schemes retain high-precision accuracy.

Proposed methods are computationally efficient, comparable to fully low-precision schemes.

Numerical experiments confirm the effectiveness of the schemes.

Abstract

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a na\"ive mixed-precision implementation of any Runge-Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed-precision RKC schemes that are instead unaffected by this limiting behaviour. We present three mixed-precision schemes: a first- and a second-order RKC method, and a first-order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first- and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method.