Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Abstract

Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter ε∈(0,1], which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. 0<ε≪1, the solution of the NKGE propagates waves with wavelength at O(1) and O(ε2) in space and time, respectively, which brings significantly numerical burdens in designing numerical methods. We compare systematically spatial/temporal efficiency and accuracy as well as ε-resolution (or ε-scalability) of different numerical methods including finite difference time domain methods, time-splitting method, exponential wave integrator, limit integrator, multiscale time integrator, two-scale formulation method and iterative exponential integrator. Finally, we adopt the multiscale time integrator to study the convergence rates from the NKGE to its limiting models when ε→0+.