A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions

TL;DR

This paper introduces a new multiscale time integrator Fourier pseudospectral method for the nonlinear Klein-Gordon equation that achieves uniform accuracy in the nonrelativistic regime, effectively handling high oscillations and large time steps.

Contribution

The paper develops a simplified multiscale time integrator with uniform accuracy for the NKGE, incorporating a multiscale decomposition and exponential wave integrator, with proven error bounds and super resolution properties.

Findings

Achieves first-order uniform accuracy in time for NKGE in the nonrelativistic regime.

Demonstrates super resolution in time, allowing larger time steps than the oscillation wavelength.

Validates theoretical error bounds through extensive numerical experiments.

Abstract

We propose a new and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter epsilon in (0,1] inversely proportional to the speed of light, and establish its uniform first-order accuracy in time in the nonrelativistic regime, i.e. 0 < epsilon << 1. In this regime, the solution of the NKGE is highly oscillatory in time with O(epsilon^2)-wavelength, which brings significant difficulties in designing uniformly accurate numerical methods. The MTI-FP is based on (i) a multiscale decomposition by frequency of the NKGE in each time interval with simplified transmission conditions, and (ii) an exponential wave integrator for temporal discretization and a Fourier pseudospectral method for spatial discretization. By adapting the energy method and the mathematical induction, we obtain two error bounds in H1-norm at O(h^{m0}+tau^2/epsilon^2) and O(h^{m0}+tau+epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution, which immediately implies a uniformly accurate error bound O(h^{m0}+tau) with respect to epsilon in (0,1]. In addition, by adopting a linear interpolation of the micro variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any time t larger than zero. Thus the proposed MTI-FP method has a super resolution property in time in terms of the Shannon sampling theory, i.e. accurate numerical solutions can be obtained even when the time step is much bigger than the O(epsilon^2)-wavelength. Extensive numerical results are reported to confirm our error bounds and demonstrate their super resolution in time. Finally the proposed MTI-FP method is applied to study numerically convergence rates of the NKGE to its different limiting models in the nonrelativistic regime.