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 the ZS and/or the incompatibility of the initial data, which is characterized by two parameters $\alpha\ge 0$ and $\beta\ge -1$. Specifically, the solution propagates waves with wavelength of $O(\varepsilon)$ and $O(1)$ in time and space, respectively, and amplitude at $O(\varepsilon^{\min\{2,\alpha,1+\beta\}})$. 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 the ZS into an asymptotic consistent formulation and adopting an integral approximation of the oscillatory term. By applying the energy method and using the limiting equation via a nonlinear Schrödinger equation with an oscillatory potential, we rigorously establish two independent error bounds at $O(h^2+\tau^2/\varepsilon)$ and $O(h^2+\tau^2+\tau\varepsilon^{\alpha^*}+\varepsilon^{1+\alpha^*})$, respectively, with $h$ the mesh size, $\tau$ the time step and $\alpha^*=\min\{1,\alpha,1+\beta\}$. Thus we obtain error bounds at $O(h^2+\tau^{4/3})$ and $O(h^2+\tau^{1+{\alpha^*}/({2+\alpha^*})})$ for well-prepared ($\alpha^*=1$) and ill-prepared ($0\le\alpha^*<1$) initial data, respectively, which are uniform in both space and time for $0<\varepsilon\le 1$ and optimal at the second order in space. Other techniques in the analysis include the cut-off technique for treating the nonlinearity and inverse estimates to bound the numerical solution. Numerical results are reported to demonstrate our error bounds.