We propose, analyze and numerically validate a new energy dissipative scheme for the Ginzburg–Landau equation by using the invariant energy quadratization approach. First, the Ginzburg–Landau equation is transformed into an equivalent formulation which possesses the quadratic energy dissipation law. After the space-discretization of the Fourier pseudo-spectral method, the semi-discrete system is proved to be energy dissipative. Using diagonally implicit Runge–Kutta scheme, the semi-discrete system is integrated in the time direction. Then the presented full-discrete scheme preserves the energy dissipation, which is beneficial to the numerical stability in long-time simulations. Several numerical experiments are provided to illustrate the effectiveness of the proposed scheme and verify the theoretical analysis.

中文翻译：

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交通流模型的四阶耗能方案

我们使用不变能量平方方法，为Ginzburg-Landau方程提出了一种新的耗能方案，对其进行了分析和数值验证。首先，将Ginzburg-Landau方程转换为具有二次能量耗散定律的等效公式。经过傅里叶伪谱方法的空间离散后，半离散系统被证明是耗能的。使用对角隐式Runge–Kutta方案，在时间方向上集成了半离散系统。然后提出的全离散方案保留了能量耗散，这有利于长时间仿真中的数值稳定性。提供了一些数值实验来说明所提方案的有效性并验证理论分析。