We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

中文翻译：

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复杂修正 KORTEWEG-DE VRIES 方程的线性隐式保能傅里叶赝谱方案

我们基于不变能量二次化方法，为复杂的修正 Korteweg-de Vries 方程提出了两种线性隐式能量保持方案。首先，引入一个新变量，并为这个方程构造一个新的哈密顿系统。空间离散采用傅里叶伪谱方法，时间离散采用Crank-Nicolson跳蛙方案。所提出的方案是线性隐式的，只需要在每个时间步求解一个线性系统。可以显示完全离散方案在离散设置中同时保存质量和能量。还给出了一些数值例子来验证所提出方案的有效性。