Abstract
In this paper, we design an energy conserving local discontinuous Galerkin (LDG) method to solve the quantum Zakharov system (QZS). We first give theoretical proof for the conservation of mass and energy and then focus our discussion on the priori estimates for the semi-discrete approximation of the QZS. With the proof for the L2 boundedness of the numerical solutions and the discrete Poincaré inequalities, we obtain the optimal error estimates. Next, we develop a decoupled and implicit linear fully-discrete scheme using the semi-implicit spectral deferred correction (SISDC) method for temporal discretization. Another strength of our approach is the arbitrary high accuracy which depends only on the degree of the approximating piecewise polynomials. Finally, several numerical examples are reported to validate it and the conservation properties. Especially, the second-order convergence rate of the QZS to its limiting model in the semi-classical limit is achieved as expected.
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