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

TL;DR

This paper introduces a novel high-order structure-preserving numerical method for the quantum Zakharov system, combining energy reformulation, symplectic integrators, and spectral methods to ensure conservation laws and high accuracy.

Contribution

The paper develops a new high-order structure-preserving method for the quantum Zakharov system using energy reformulation and symplectic Runge-Kutta, ensuring conservation of mass and Hamiltonian energy.

Findings

Achieves arbitrary high-order accuracy in time.

Preserves discrete mass and Hamiltonian energy exactly.

Demonstrates efficiency through numerical examples.

Abstract

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: (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; (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 and can preserve the discrete mass and original Hamiltonian energy exactly. 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.