Metastable energy strata in numerical discretizations of weakly nonlinear wave equations

TL;DR

This paper demonstrates that symplectic trigonometric integrators can accurately reproduce metastable energy strata in numerical simulations of weakly nonlinear wave equations over long time periods.

Contribution

It shows that symplectic trigonometric integrators correctly reproduce metastable energy strata in weakly nonlinear wave equations over long times.

Findings

Metastable energy strata are preserved by symplectic integrators.

Long-time qualitative accuracy of energy distribution is achieved.

Numerical methods can replicate theoretical metastable behaviors.

Abstract

The quadratic nonlinear wave equation on a one-dimensional torus with small initial values located in a single Fourier mode is considered. In this situation, the formation of metastable energy strata has recently been described and their long-time stability has been shown. The topic of the present paper is the correct reproduction of these metastable energy strata by a numerical method. For symplectic trigonometric integrators applied to the equation, it is shown that these energy strata are reproduced even on long time intervals in a qualitatively correct way.