Error Analysis of Trigonometric Integrators for Semilinear Wave Equations

Abstract

An error analysis of trigonometric integrators (or exponential integrators) applied to spatial semi-discretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given. In particular, optimal second-order convergence is shown requiring only that the exact solution is of finite energy. The analysis is uniform in the spatial discretization parameter. It covers the impulse method which coincides with the method of Deuflhard and the mollified impulse method of García-Archilla, Sanz-Serna & Skeel as well as the trigonometric methods proposed by Hairer & Lubich and by Grimm & Hochbruck. The analysis can also be used to explain the convergence behaviour of the Störmer-Verlet/leapfrog discretization in time. (2010): 65M15, 65P10, 65L70, 65M20. Mathematics Subject Classification

Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation

“…We split the error formula into a quadrature error and several remainder terms as in e.g. [4] or [7]. The main novelty of our approach is the use of fractional convergence results.…”

Functionally-fitted energy-preserving integrators for Poisson systems

“…Two important propositions of the nonlinearity f (y) given in [9] are needed in this paper and we summarize them as follows.…”

Adiabatic midpoint rule for the dispersion-managed nonlinear Schrödinger equation

“…in the spirit of (11) and (16). For simplicity, we fix m ∈ Z, (j, k, l) ∈ I m , (p, q, r) ∈ I j , and write for short ω = ω [jklm] , ω = ω [pqrj] and in particular…”

“…In the following, we mention only references where partial differential equations are considered. Trigonometric integrators for semilinear wave equations have been proposed and analyzed, e.g., in [16,19,21]. Special methods for linear Schrödinger equations in the semiclassical regime have been developed, e.g., in [5,13] and references therein.…”