Simulation of Ginzburg–Landau equation via rational RBF partition of unity approach

In the recent decade, several numerical procedures are developed based on the radial basis functions (RBFs) in the strong and the weak form of the mathematical model. These numerical algorithms have been used to solve a wide range of differential equations. However, the accuracy of RBFs collocation method for some partial differential equations (PDEs) is low thus a modification of RBFs collocation plan is required. The main aim of the current paper is to employ an improvement of RBFs collocation algorithm i.e. rational RBFs (RRBFs) collocation method based on the partition of unity (PU) idea to get the numerical solution of the multi-dimensional Ginzburg–Landau equation. It is clear that the RBFs collocation approach is an important numerical procedure for solving PDEs in non-rectangular physical regions. For differential equations with sufficiently smooth solutions, the RBF collocation technique generates acceptable and efficient accuracy. The RBF collocation method may produce solutions with non-physical oscillations for the underlying functions which have steep gradients or discontinuities. Using a fourth-order time-split approach and rational RBFs (RRBFs) collocation method, a new numerical procedure is proposed. First, a fourth-order time-split approach is used to discrete the time variable. Then, a combination of RBFs-PU collocation technique has been developed to get a full-discrete scheme. At the end, several examples have been studied to show the stability, convergence, and accuracy of the proposed numerical algorithm.