Metastable energy strata in numerical discretizations of weakly nonlinear wave equations
For researchers in numerical analysis and wave dynamics, this work validates that symplectic integrators preserve long-time qualitative behavior of energy strata, though it is an incremental extension of known stability results.
The paper shows that symplectic trigonometric integrators correctly reproduce metastable energy strata in numerical discretizations of weakly nonlinear wave equations over long time intervals, confirming qualitative agreement with theoretical predictions.
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.