A uniformly and optimally accurate multiscale time integrator method for the Klein-Gordon-Zakharov system in the subsonic limit regime

We present a uniformly and optimally accurate numerical method for discretizing the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter 0<ε≤1, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0<ε≪1, the solution of KGZ system propagates waves with O(ε)- and O(1)-wavelength in time and space, respectively, and rapid outspreading initial layers with speed O(1∕ε) in space due to the singular perturbation of the wave operator in KGZ and/or the incompatibility of the initial data. Based on a multiscale decomposition by frequency and amplitude, we propose a multiscale time integrator Fourier pseudospectral method by applying the Fourier spectral discretization for spatial derivatives followed by using the exponential wave integrator in phase space for integrating the decomposed system at each time step. The method is explicit and easy to be implemented. Extensive numerical results show that the MTI-FP method converges optimally in both space and time, with exponential and quadratic convergence rate, respectively, which is uniformly for ε∈(0,1]. Finally, the method is applied to study the convergence rates of the KGZ system to its limiting models in the subsonic limit and wave dynamics and interactions of the KGZ system in 2D.