The fourth-order time-discrete scheme and split-step direct meshless finite volume method for solving cubic–quintic complex Ginzburg–Landau equations on complicated geometries

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.