Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations
D. Cohen, E. Hairer and Ch. Lubich
Abstract. For classes of symplectic and symmetric time-stepping methods - trigonometric integrators and the Stšrmer-Verlet or leapfrog method - applied to spectral semi-discretizations of semilinear wave equations in a weakly nonlinear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.
Key Words. Nonlinear wave equation, conservation of energy, momentum and actions, long-time behaviour, trigonometric methods, leapfrog, Stšrmer--Verlet method.