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We establish uniform error bounds of a finite difference method for theKlein-Gordon-Zakharov system (KGZ) with a dimensionless parameter $\varepsilon\in (0,1]$, which is inversely proportional to the acoustic speed. In thesubsonic limit regime, i.e. $0<\varepsilon \ll 1$, the solution propagateshighly oscillatory waves in time and/or rapid outgoing initial layers in spacedue to the singular perturbation in the Zakharov equation and/or theincompatibility of the initial data. Specifically, the solution propagateswaves with $O(\varepsilon)$-wavelength in time and $O(1)$-wavelength in spaceas well as outgoing initial layers in space at speed $O(1/\varepsilon)$. Thishigh oscillation in time and rapid outgoing waves in space of the solutioncause significant burdens in designing numerical methods and establishing errorestimates for KGZ. By adapting an asymptotic consistent formulation, we proposea uniformly accurate finite difference method and rigorously establish twoindependent error bounds at $O(h^2+\tau^2/\varepsilon)$ and$O(h^2+\tau+\varepsilon)$ with $h$ mesh size and $\tau$ time step. Thus weobtain a uniform error bound at $O(h^2+\tau)$ for $0<\varepsilon\le 1$. Themain techniques in the analysis include the energy method, cut-off of thenonlinearity to bound the numerical solution, the integral approximation of theoscillatory term, and $\varepsilon$-dependent error bounds between thesolutions of KGZ and its limiting model when $\varepsilon\to0^+$. Finally,numerical results are reported to confirm our error bounds.

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