A Uniformly Accurate Multiscale Time Integrator for the Klein-Gordon-Schr\"odinger Equations in the Nonrelativistic Regime via Simplified Transmission Conditions

TL;DR

This paper introduces a new multiscale time integrator Fourier pseudospectral method for Klein-Gordon-Schrödinger equations that achieves uniform accuracy across regimes, effectively handling oscillations as epsilon approaches zero.

Contribution

The paper develops a simplified, uniformly accurate multiscale time integrator method with rigorous error bounds, enabling efficient simulation of KGS equations in the nonrelativistic regime.

Findings

The method achieves first-order uniform accuracy in time for epsilon in (0,1].

Error bounds are established in H^1-norm showing optimal convergence.

Numerical experiments confirm the super-resolution property and validate theoretical error estimates.

Abstract

We propose a novel and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Klein-Gordon-Schr\"odinger (KGS) equations with a dimensionless parameter epsilon in (0,1], where epsilon is inversely proportional to the speed of light. The proposed MTI-FP method is rigorously proved to achieve uniform first-order accuracy in time in the nonrelativistic regime, i.e., as epsilon->0. In this regime, the solution of the KGS equations exhibits temporal oscillations with an O(epsilon^2)-wavelength, imposing stringent resolution requirements on classical numerical methods. The uniformly accurate MTI-FP method is built upon two key points: (i) a multiscale decomposition by frequency in each time interval with simplified transmission conditions, and (ii) an exponential integrator for temporal discretization combined with the Fourier pseudospectral method for spatial discretization. Using the energy method and mathematical induction, we rigorously establish two independent error bounds in H^1-norm at O(h^{m0-1} + tau^2/epsilon^2) and O(h^{m0-1} + epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution. These estimates imply that the MTI-FP method converges uniformly and optimally in space, and uniformly in time at O(tau) with respect to epsilon in (0,1]. Furthermore, by incorporating a linear interpolation of the micro-variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any t>0. Consequently, the proposed MTI-FP method has a super-resolution property in time from the perspective of Shannon sampling theory. Ample numerical experiments are provided to validate the error estimates and to demonstrate the super-resolution property. Finally, the method is applied to numerically investigate the convergence rates of the KGS equations to different limiting models.