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

Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\epsilon \in (0,1]$, which is inversely proportional to the speed of light. In the nonrelativestic limit regime, i.e. $0<\epsilon \ll 1$, the solution of the NKGE propagates waves with wavelength at $O(1)$ and $O({\epsilon }^2)$ 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 $\epsilon$-resolution (or $\epsilon$-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 $\epsilon \to 0^{+}$.