Our motivation in this contribution is to propose a new numerical algorithm for solving cubic–quintic complex Ginzburg-Landau (CQCGL) equations. The developed technique is based on the following stages. At the first step, the nonlinear CQCGL equation is splitted in the three problems that two of them don’t have the space derivative e.g problems (I) and (III) and one of them has the space derivative e.g Problem (II). At the second stage, the Problems (I) and (III) can be considered as two ODEs and they are solved by using a fourth-order exponential time differencing Runge-Kutta (ETDRK4) method to get a high-order numerical approximation. Furthermore, the Problem (II) is solved by using direct meshless finite volume method. The proposed method is a new high-order numerical procedure based on a truly meshless method for solving the complex PDEs on non-rectangular computational domains. Moreover, various samples are investigated that verify the efficiency of the new numerical scheme.

中文翻译：

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求解复杂几何上三次-五次复Ginzburg-Landau方程的四阶时间离散格式和分步直接无网格有限体积法

我们在此贡献中的动机是提出一种新的数值算法来求解三次五次复金茨堡-朗道 (CQCGL) 方程。开发的技术基于以下阶段。第一步，非线性CQCGL方程被分解为三个问题，其中两个没有空间导数，例如问题（I）和（III），其中一个有空间导数，例如问题（II）。在第二阶段，问题 (I) 和 (III) 可以被视为两个 ODE，并使用四阶指数时间差分 Runge-Kutta (ETDRK4) 方法求解，以获得高阶数值近似。此外，问题（II）是通过使用直接无网格有限体积法来解决的。所提出的方法是一种新的高阶数值程序，它基于真正的无网格方法，用于求解非矩形计算域上的复杂 PDE。此外，还研究了各种样本，以验证新数值方案的效率。