Arbitrary high-order structure-preserving methods for the quantum Zakharov system

In this paper, we present a new methodology to develop arbitrary high-order structure-preserving methods for solving the quantum Zakharov system. The key ingredients of our method are as follows: (i) the original Hamiltonian energy is reformulated into a quadratic form by introducing a new quadratic auxiliary variable; (ii) based on the energy variational principle, the original system is then rewritten into a new equivalent system which inherits the mass conservation law and a quadratic energy; and (iii) the resulting system is discretized by symplectic Runge-Kutta method in time combining with the Fourier pseudo-spectral method in space. The proposed method achieves arbitrary high-order accurate in time in a periodic domain and can exactly preserve the discrete mass and original Hamiltonian energy. Moreover, an efficient iterative solver is presented to solve the resulting discrete nonlinear equations. Finally, ample numerical examples are presented to demonstrate the theoretical claims and illustrate the efficiency of our methods.

Automated summary

This paper presents a new methodology for developing arbitrary high-order structure-preserving methods to solve the quantum Zakharov system. The method reformulates the original Hamiltonian energy into a quadratic form by introducing a new quadratic auxiliary variable, and then rewrites the original system into a new equivalent system based on the energy variational principle. The proposed method achieves arbitrary high-order accuracy in time in a periodic domain and exactly preserves the discrete mass and original Hamiltonian energy.