A uniformly accurate (UA) multiscale time integrator Fourier pseoduspectral method for the Klein-Gordon-Schrodinger equations in the nonrelativistic limit regime

TL;DR

This paper introduces a multiscale time integrator Fourier pseudospectral method for efficiently solving Klein-Gordon-Schrodinger equations in the nonrelativistic limit, achieving uniform accuracy and optimal convergence rates.

Contribution

The paper develops a novel multiscale time integrator Fourier pseudospectral method with rigorous error analysis for Klein-Gordon-Schrodinger equations in the nonrelativistic limit.

Findings

Achieves uniform convergence in space and time with optimal rates.

Requires meshing strategy with fixed time and space steps regardless of .

Numerical results confirm the sharpness of error bounds.

Abstract

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein-Gordon-Schr\"{o}dinger (KGS) equations in the nonrelativistic limit regime with a dimensionless parameter $0<\varepsilon\le1$ which is inversely proportional to the speed of light. In fact, the solution to the KGS equations propagates waves with wavelength at $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively, when $0<\varepsilon\ll 1$, which brings significantly numerical burdens in practical computation. The MTI-FP method is designed by adapting a multiscale decomposition by frequency to the solution at each time step and applying the Fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives, respectively. We rigorously establish two independent error bounds for the MTI-FP at $O(\tau^2/\varepsilon^2+h^{m_0})$ and $O(\varepsilon^2+h^{m_0})$ for $\varepsilon\in(0,1]$ with $\tau$ time step size, $h$ mesh size and $m_0\ge 4$ an integer depending on the regularity of the solution, which imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regime when either $\varepsilon=O(1)$ or $0<\varepsilon\le \tau$. Thus the meshing strategy requirement (or $\varepsilon$-scalability) of the MTI-FP is $\tau=O(1)$ and $h=O(1)$ for $0<\varepsilon\ll 1$, which is significantly better than classical methods. Numerical results demonstrate that our error bounds are optimal and sharp. Finally, the MTI-FP method is applied to study numerically convergence rates of the KGS equations to the limiting models in the nonrelativistic limit regime.