A uniformly and optimally accurate method for the Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime

TL;DR

This paper introduces a multiscale numerical method for the Klein-Gordon-Zakharov system that remains accurate across different parameter regimes, effectively capturing wave propagation and initial layers, and analyzing convergence to limiting models.

Contribution

A multiscale time integrator Fourier pseudospectral method is developed for the KGZ system, achieving uniform accuracy in high-frequency and subsonic limits, and enabling convergence analysis.

Findings

Method is explicit, efficient, and uniformly accurate for all parameters.

Numerical results confirm high accuracy and efficiency.

Convergence rates to limiting models are investigated.

Abstract

We present a uniformly and optimally accurate numerical method for solving the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters $0<\epsilon\le1$ and $0<\gamma\le 1$, which are inversely proportional to the plasma frequency and the acoustic speed, respectively. In the simultaneous high-plasma-frequency and subsonic limit regime, i.e. $\epsilon<\gamma\to 0^+$, the KGZ system collapses to a cubic Schr\"odinger equation, and the solution propagates waves with $O(\epsilon^2)$-wavelength in time and meanwhile contains rapid outgoing initial layers with speed $O(1/\gamma)$ in space due to the incompatibility of the initial data. By presenting a multiscale decomposition of the KGZ system, we propose a multiscale time integrator Fourier pseduospectral method which is explicit, efficient and uniformly accurate for solving the KGZ system for all $0<\epsilon<\gamma\leq1$. Numerical results are reported to show the efficiency and accuracy of scheme. Finally, the method is applied to investigate the convergence rates of the KGZ system to its limiting models when $\epsilon<\gamma\to 0^+$.