Stability and convergence of trigonometric integrator pseudospectral discretization for N-coupled nonlinear Klein–Gordon equations

This work serves as an improvement on a recent paper (Dong, 2013) [9], in which the N-coupled nonlinear Klein–Gordon equations were solved numerically by a fully explicit trigonometric integrator Fourier pseudospectral (TIFP) method. This TIFP method is second-order accurate in time and spectral-order accurate in space; however, in the previous work there was an absence of rigorous stability and convergence analysis. Moreover, numerical studies in this work suggest that this TIFP method suffers from a stability condition τ=O(h) (τ and h refer to time step and space mesh size). To relax such a restriction while keeping the convergence properties and explicitness, we propose two modified TIFP methods, motivated by the mollified impulse and Gautschi-type integrators for oscillatory ODEs. For the modifications considered here, linear stability and rigorous error estimates in the energy space are carried out, which are the main achievements gained in this work. Meanwhile, numerical results are also presented. Ideas of this work also suggest a general framework for proposing and analyzing efficient numerical methods for coupled wave-type equations.