Stability Analysis and Performance Evaluation of Mixed-Precision Runge-Kutta Methods

TL;DR

This paper analyzes the stability and performance of mixed-precision Runge-Kutta methods for solving differential equations, demonstrating their potential for improved efficiency without sacrificing accuracy.

Contribution

It provides a detailed stability analysis and performance evaluation of newly proposed mixed-precision Runge-Kutta methods, including implementation and empirical results.

Findings

Mixed-precision methods maintain accuracy with reduced runtime.

Stability is sensitive to low-precision rounding errors.

Significant runtime reductions achieved for given accuracy levels.

Abstract

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in 4. These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations. In this work we analyze the stability properties of these methods and their sensitivity to the low precision rounding errors, and demonstrate their performance in terms of accuracy and efficiency. We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed precision additive Runge-Kutta (MP-ARK) methods. The convergence, accuracy, runtime, and energy consumption of these methods is explored. We show that for a given level of accuracy, suitably chosen MP-ARK methods may provide significant reductions in runtime.