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.
翻译:在本文中,我们提出了一个解决Zakharov-Rubenchik方程式的高序节能计划的新颖类别,其主要设想是首先引入一个二次辅助变量,将汉密尔顿能源转化为经修改的二次能源,然后将原系统重新改制成一个能满足质量、经修改的能量和两个线性变异物的等效系统,同时采用静电龙格-库塔方法以及Fourier模拟光谱方法来计算重新开发的系统的解决办法。拟议计划的主要好处是,它可以在时间上达到任意高序精确度,并保护三种变异物:质量、汉密尔密尔顿能源和两种线性变异物。此外,还提议一个高效的固定点循环,以解决由此产生的拟议方案的非线性方程式。一些实验旨在验证理论结果。