Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime

Abstract:

We rigorously analyze the error estimates of the conservative finite difference method (CNFD) for the Zakharov system (ZS) with a dimensionless parameter $\varepsilon \in (0,1]$, which is inversely proportional to the ion acoustic speed. When $\varepsilon \to 0^+$, ZS collapses to the standard nonlinear Schrödinger equation (NLS). In the subsonic limit regime, i.e., $\varepsilon \to 0^+$, there exist highly oscillatory initial layers in the solution. The initial layers propagate with $O(\varepsilon )$ wavelength in time, $O(1)$ and $O(\varepsilon ^2)$ amplitudes, for the ill-prepared initial data and well-prepared initial data, respectively. This oscillatory behavior brings significant difficulties in analyzing the errors of numerical methods for solving the Zakharov system. In this work, we show the CNFD possesses the error bounds $h^2/\varepsilon +\tau ^2/\varepsilon ^3$ in the energy norm for the ill-prepared initial data, where $h$ is mesh size and $\tau$ is time step. For the well-prepared initial data, CNFD is uniformly convergent for $\varepsilon \in (0,1]$, with second-order accuracy in space and $O(\tau ^{4/3})$ accuracy in time. The main tools involved in the analysis include cut-off technique, energy methods, $\varepsilon$-dependent error estimates of the ZS, and $\varepsilon$-dependent error bounds between the numerical approximate solution of the ZS and the solution of the limit NLS. Our approach works in one, two and three dimensions, and can be easily extended to the generalized Zakharov system and nonconservative schemes. Numerical results suggest that the error bounds are sharp for the plasma densities and the error bounds of the CNFD for the electric fields are the same as those of the splitting methods.