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

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) the original system is then rewritten into a new equivalent system which inherits the quadratic energy based on the energy variational principle; (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 Hamiltonian energy exactly. Moreover, an efficient iterative solver is presented to solve the resulting discrete nonlinear equations. Finally, extensive numerical examples are provided to demonstrate the theoretical claims and illustrate the efficiency of our method.

Arbitrary High-Order Structure-Preserving Schemes for Generalized Rosenau-Type Equations

“…[13, Proposition 7.1.1]. Therefore, motivated by [11,50], we propose an efficient fixedpoint iteration solver for the nonlinear equations in the numerical schemes under consideration. For simplicity, we consider the 4-th MP scheme with the RK coefficients given in Table 1.…”

Arbitrary high-order structure-preserving schemes for the generalized Rosenau-type equation

“…In this section, motivated by [6,32], we propose an efficient fixed pointed iteration solver for the nonlinear equations of the proposed schemes. For simplicity, we consider the 4th-MP scheme where the RK coefficient is given in Table 1.…”

Arbitrarily high-order energy-preserving schemes for the Zakharov-Rubenchik equation

“…Inspired by [9,36], we propose an efficient fixed pointed iteration solver for solving the nonlinear equations of the Scheme 3.1 in this section. For convenience, we only take the proposed FPRK-2 scheme into consideration, in which the RK coefficients are presented in Table 1.…”