ON ENERGY LAWS AND STABILITY OF RUNGE-KUTTA METHODS FOR LINEAR SEMINEGATIVE PROBLEMS

Sun，Zheng1; Wei，Yuanzhe2; Wu，Kailiang3

2022

10.1137/22M1472218

SIAM JOURNAL ON NUMERICAL ANALYSIS   Impact Factor & Quartile

0036-1429

1095-7170

This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge-Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of only explicit RK methods in [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal., 57 (2019), pp. 1158-1182]. The established energy identities provide a precise characterization on whether and how the energy dissipates in the RK discretization, thereby leading to weak and strong stability criteria of RK methods. Furthermore, we discover a unified energy identity for all the diagonal Padé approximations, based on an analytical Cholesky type decomposition of a class of symmetric matrices. The structure of the matrices is very complicated, rendering the discovery of the unified energy identity and the proof of the decomposition highly challenging. Our proofs involve the construction of technical combinatorial identities and novel techniques from the theory of hypergeometric series. Our framework is motivated by a discrete analogue of integration by parts technique and a series expansion of the continuous energy law. In some special cases, our analyses establish a close connection between the continuous and discrete energy laws, enhancing our understanding of their intrinsic mechanisms. Several specific examples of implicit methods are given to illustrate the discrete energy laws. A few numerical examples further confirm the theoretical properties.

energy laws energy method L2-stability Padé approximations Runge-Kutta methods

SCI

English

Others

National Natural Science Foundation of China[12171227];Norsk Sykepleierforbund[DMS-2208391];

Mathematics

Mathematics, Applied

WOS:000889757900004

SIAM PUBLICATIONS

MATHEMATICS

2-s2.0-85138470113

Scopus

1.Department of Mathematics,The University of Alabama,Tuscaloosa,35487,United States
2.Department of Mathematics,Southern University of Science and Technology,Shenzhen,Guangdong,518055,China
3.Department of Mathematics,SUSTech International Center for Mathematics,Southern University of Science and Technology,National Center for Applied Mathematics Shenzhen (NCAMS),Shenzhen,Guangdong,518055,China

Sun，Zheng,Wei，Yuanzhe,Wu，Kailiang. ON ENERGY LAWS AND STABILITY OF RUNGE-KUTTA METHODS FOR LINEAR SEMINEGATIVE PROBLEMS[J]. SIAM JOURNAL ON NUMERICAL ANALYSIS,2022,60(5):2448-2481.

Sun，Zheng,Wei，Yuanzhe,&Wu，Kailiang.(2022).ON ENERGY LAWS AND STABILITY OF RUNGE-KUTTA METHODS FOR LINEAR SEMINEGATIVE PROBLEMS.SIAM JOURNAL ON NUMERICAL ANALYSIS,60(5),2448-2481.

Sun，Zheng,et al."ON ENERGY LAWS AND STABILITY OF RUNGE-KUTTA METHODS FOR LINEAR SEMINEGATIVE PROBLEMS".SIAM JOURNAL ON NUMERICAL ANALYSIS 60.5(2022):2448-2481.