Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime

Abstract

We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter $\varepsilon \in (0,1]$, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. $0<\varepsilon \ll 1$, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with $O(\varepsilon)$-wavelength in time and $O(1)$-wavelength in space as well as outgoing initial layers in space at speed $O(1/\varepsilon)$. This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ. By adapting an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at $O(h^2+τ^2/\varepsilon)$ and $O(h^2+τ+\varepsilon)$ with $h$ mesh size and $τ$ time step. Thus we obtain a uniform error bound at $O(h^2+τ)$ for $0<\varepsilon\le 1$. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and $\varepsilon$-dependent error bounds between the solutions of KGZ and its limiting model when $\varepsilon\to0^+$. Finally, numerical results are reported to confirm our error bounds.