Uniform error bound of a Crank–Nicolson‐type finite difference scheme for Zakharov system in the subsonic limit regime

Abstract

In this paper, we use the order reduction method to present a Crank–Nicolson‐type finite difference scheme for Zakharov system (ZS) with a dimensionless parameter ε∈false(0,1false]$$ \varepsilon \in \left(0,1\right] $$, which is inversely proportional to the ion acoustic speed. The proposed scheme is proved to perfectly inherit the mass and energy conservation possessed by ZS, while the invariants satisfied by most existing schemes are expressed by two‐level's solution at each time step. In the subsonic limit regime, that is, when 0<ε≪1$$ 0&lt;\varepsilon \ll 1 $$, the solution of ZS propagates rapidly oscillatory initial layers in time, and this brings significant difficulties in designing numerical methods and establishing the error estimates, especially in the subsonic limit regime. After proving the solvability of the proposed scheme, we use the cut‐off function technique and energy method to rigorously analyze two independent error estimates for the well‐prepared, less‐ill‐prepared, ill‐prepared initial data, respectively, which are uniform in both time and space for ε∈false(0,1false]$$ \varepsilon \in \left(0,1\right] $$ and optimal at second order in space. Numerical examples are carried out to verify the theoretical results and show the effectiveness of the proposed scheme.