A uniformly accurate multiscale time integrator spectral method for the Klein-Gordon-Zakharov system in the high-plasma-frequency limit regime

A multiscale time integrator sine pseudospectral (MTI-SP) method is presented for discretizing the Klein-Gordon-Zakharov (KGZ) system with a dimensionless parameter 0<ε≤1, which is inversely proportional to the plasma frequency. In the high-plasma-frequency limit regime, i.e. 0<ε≪1, the solution of the KGZ system propagates waves with amplitude at O(1) and wavelength at O(ε2) in time and O(1) in space, which causes significantly numerical burdens due to the high oscillation in time. The main idea of the numerical method is to carry out a multiscale decomposition by frequency (MDF) to the electric field component of the solution at each time step and then apply the sine pseudospectral discretization for spatial derivatives followed by using the exponential wave integrator in phase space for integrating the MDF and the equation of the ion density component. The method is explicit and easy to be implemented. Extensive numerical results show that the MTI-SP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O(τ) for ε∈(0,1] with τ time step size and optimally with quadratic convergence rate at O(τ2) in the regime when either ε=O(1) or 0<ε≤τ. Thus the meshing strategy requirement (or ε-scalability) of the MTI-SP for the KGZ system in the high-plasma-frequency limit regime is τ=O(1) and h=O(1) for 0<ε≪1, which is significantly better than classical methods in the literatures. Finally, we apply the MTI-SP method to study the convergence rates of the KGZ system to its limiting models in the high-plasma-frequency limit and the interactions of bright solitons of the KGZ system, and to identify certain parameter regimes that the solution of the KGZ system will be blow-up in one dimension.