Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation

Abstract

We present a uniformly accurate finite difference method and establish rigorously its uniform error bounds for the Zakharov system (ZS) with a dimensionless parameter $0<\varepsilon\le 1$, which is inversely proportional to the speed of sound. In the subsonic limit regime, i.e., $0<\varepsilon\ll 1$, the solution propagates highly oscillatory waves and/or rapid outgoing initial layers due to the perturbation of the wave operator in ZS and/or the incompatibility of the initial data which is characterized by two nonnegative parameters $\alpha$ and $\beta$. Specifically, the solution propagates waves with $O(\varepsilon)$- and $O(1)$-wavelength in time and space, respectively, and amplitude at $O(\varepsilon^{\min\{2,\alpha,1+\beta\}})$ and $O(\varepsilon^\alpha)$ for well-prepared ($\alpha\ge1$) and ill-prepared ($0\le \alpha<1$) initial data, respectively. This high oscillation of the solution in time brings significant difficulties in designing numerical methods and establishing their error bounds, especially in the subsonic limit regime. A uniformly accurate finite difference method is proposed by reformulating ZS into an asymptotic consistent formulation and adopting an integral approximation of the oscillatory term. By adapting the energy method and using the limiting equation via a nonlinear Schr\"{o}dinger equation with an oscillatory potential, we rigorously establish two independent error bounds and obtain error bounds at $O(h^2+\tau^{4/3})$ and $O(h^2+\tau^{1+\frac{\alpha}{2+\alpha}})$ for well-prepared and ill-prepared initial data, respectively, which are uniform in both space and time for $0<\varepsilon\le 1$ and optimal at the second order in space. Numerical results are reported to demonstrate that our error bounds are sharp.