Arbitrarily high-order energy-preserving schemes for the Zakharov-Rubenchik equations. (English) Zbl 1512.65187
In this paper, the authors present a novel class of high-order energy-preserving schemes for solving the Zakharov-Rubenchik equations. The main idea of the scheme is first to introduce an quadratic auxiliary variable to transform the Hamiltonian energy into a modified quadratic energy and the original system is then reformulated into an equivalent system which satisfies the mass, modified energy as well as two linear invariants. The symplectic Runge-Kutta method in time, together with the Fourier pseudo-spectral method in space is employed to compute the solution of the reformulated system. The main benefit of the proposed schemes is that it can achieve arbitrarily high-order accurate in time and conserve the three invariants: mass, Hamiltonian energy and two linear invariants. In addition, an efficient fixed-point iteration is proposed to solve the resulting nonlinear equations of the proposed schemes. Several experiments are addressed to validate the theoretical results.
Reviewer: Qifeng Zhang (Hangzhou)
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 47H10 | Fixed-point theorems |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
Zakharov-Rubenchik equations; energy-preserving scheme; symplectic Runge-Kutta method; quadratic auxiliary variable approach; Fourier pseudo-spectral methodSoftware:
LIMbookReferences:
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