Communications in Mathematical Sciences

Volume 18 (2020)

Number 2

A class of efficient spectral methods and error analysis for nonlinear Hamiltonian systems

Pages: 395 – 428



Jing An (Beijing Computational Science Research Center, Beijing, China; and School of Mathematical Sciences, Guizhou Normal University, Guiyang, China)

Waixiang Cao (School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Zhimin Zhang (Beijing Computational Science Research Center, Beijing, China; and Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)


We investigate efficient numerical methods for nonlinear Hamiltonian systems. Three polynomial spectral methods (including spectral Galerkin, Petrov–Galerkin, and collocation methods) coupled with domain decomposition are presented and analyzed. Our main results include the energy and symplectic structure-preserving properties and error estimates. We prove that the spectral Petrov–Galerkin method preserves the energy exactly while both the spectral Gauss collocation and spectral Galerkin methods are energy conserving up to spectral accuracy. While it is well known that collocation at Gauss points preserves symplectic structure, we prove that, for both the Petrov–Galerkin method and the spectral Galerkin method, the error in symplecticity decays with spectral accuracy. Finally, we show that all three methods converge exponentially with respect to the polynomial degree. Numerical experiments indicate that our algorithms are efficient.


nonlinear Hamiltonian system, spectral methods, error analysis, energy conservation, symplectic structure

2010 Mathematics Subject Classification

65M15, 65M70, 65N30

The authors’ research was supported in part by NSFC grants No. 11871106, No. 11726604, No. 11726603, No. 11871092, U1930402, the Fundamental Research Funds for the Central Universities No. 2017NT10, and Guizhou Provincial Education Department Foundation (Qianjiaohe No. KY[2018]041).

The full-text of this online article was revised on 21 June 2022, to indicate that Waixiang Cao is the corresponding author.

Received 11 January 2019

Accepted 19 October 2019

Published 20 June 2022