A fourth-order compact finite difference scheme for the quantum Zakharov system that perfectly inherits both mass and energy conservation

Abstract

In this paper, we propose and analyze a new fourth-order compact finite difference scheme for solving the quantum Zakharov system (QZS). The new scheme perfectly inherits the mass and energy conservation possessed by the QZS, while the energy preserved by the existing schemes expressed by two-level's solution at each time step. By using the energy method to analyze an equivalent form of the difference scheme, without any requirement on the grid ratio, we establish rigorously the optimal error estimate of the proposed scheme in the energy norm. The accuracy is shown to be fourth order in space and second order in time. Numerical examples are presented to verify the theoretical results and show the effectiveness of the proposed scheme.