Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

Authors:
David Cohen;Ernst Hairer;Christian Lubich
Affiliations:
University of Basel, Mathematisches Institut, 4051, Basel, Switzerland;University of Geneva, Dept. de Mathématiques, 1211, Geneva 4, Switzerland;University of Tübingen, Mathematisches Institut, 72076, Tübingen, Germany
Venue:
Numerische Mathematik
Year:
2008

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Abstract

For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Störmer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear 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.