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

doi:10.1007/s42967-023-00315-4

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 705-738.doi: 10.1007/s42967-023-00315-4

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

Ben Burnett¹, Sigal Gottlieb¹, Zachary J. Grant²

1. Center for Scientific Computing and Data Science Research, UMass Dartmouth, North Dartmouth, Massachusetts, USA;
2. CSME, Michigan State University, East Lansing, Michigan, USA

Received:2022-12-16 Revised:2023-09-07 Published:2024-04-16
Contact: Sigal Gottlieb,E-mail:sgottlieb@umassd.edu;Ben Burnett,E-mail:bburnett@umassd.edu;Zachary J. Grant,E-mail:zack.j.grant@gmail.com E-mail:sgottlieb@umassd.edu;bburnett@umassd.edu;zack.j.grant@gmail.com
Supported by:
Burnett and Gottlieb’s work was partially supported by ONR UMass Dartmouth Marine and UnderSea Technology (MUST) grant N00014-20-1-2849 under the project S31320000049160, by DOE grant DE-SC0023164 sub-award RC114586-UMD, and by AFOSR grants FA9550-18-1-0383 and FA9550-23-1-0037. Grant’s work was partially supported by Michigan State University, by AFOSR grants FA9550-19-1-0281 and FA9550-18-1-0383 and by DOE grant DE-SC0023164.

Abstract

Abstract: Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed. 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, and runtime of these methods are explored. We show that for a given level of accuracy, suitably chosen MP-ARK methods may provide significant reductions in runtime.

Key words: Mixed precision, Runge-Kutta methods, Additive methods, Accuracy

Ben Burnett, Sigal Gottlieb, Zachary J. Grant. Stability Analysis and Performance Evaluation of Additive Mixed-Precision Runge-Kutta Methods[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 705-738.

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

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