Arbitrarily High-Order Energy-Preserving Schemes for the Zakharov-Rubenchik Equations

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.

Automated summary

In this paper, a novel class of high-order energy-preserving schemes for solving the Zakharov-Rubenchik equations is proposed. The schemes introduce a quadratic auxiliary variable to transform the Hamiltonian energy and reformulate the original system into an equivalent system satisfying multiple invariants. The schemes achieve high-order accuracy in time and conserve the mass, Hamiltonian energy, and two linear invariants.