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

TL;DR

This paper introduces high-order energy-preserving numerical schemes for the Zakharov-Rubenchik equations that conserve multiple invariants and achieve high accuracy through a novel reformulation and spectral methods.

Contribution

A new class of arbitrarily high-order energy-preserving schemes using auxiliary variables and symplectic Runge-Kutta methods for Zakharov-Rubenchik equations.

Findings

Schemes conserve mass, energy, and invariants.

Achieve high-order temporal accuracy.

Validated through numerical experiments.

Abstract

In this paper, we 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.