The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods

In the current work, we consider the two-dimensional time-dependent phase field crystal (PFC) and the modified phase field crystal models (MPFC) to obtain their numerical solutions. For this purpose, we apply two numerical meshless methods based on radial basis functions (RBFs) and also two meshless methods which are based on moving least squares method (MLS). Four techniques developed in this paper are: globally radial basis functions (GRBFs), radial basis functions pseudo-spectral (RBFs-PS), moving least squares (MLS) and generalized moving least squares (GMLS) approximations. Two methods based on RBFs are global and the other methods based on moving least squares are local procedures. As is well-known, the meshless methods are suitable techniques for the numerical solution of partial differential equations on regular and non-regular domains with different choices of grids in high-dimensions. Applying the new methods on spatial domain and also using the semi-implicit scheme for time variable, yield a linear system of algebraic equations. To solve this linear system, the LU decomposition algorithm and command “backslash” in MATLAB software (for GMLS method) are applied. The implementation of boundary conditions in the RBFs-PS is applied directly, because the boundary conditions of the mentioned problems are periodic. Some numerical results show that the obtained simulations via four proposed methods are acceptable for approximating the solution of models investigated in the current paper. Moreover in the Appendix of the paper, a MATLAB code for GMLS method is written.