Numerical simulations of nonlocal phase-field and hyperbolic nonlocal phase-field models via localized radial basis functions-based pseudo-spectral method (LRBF-PSM)

In this paper we consider the two-dimensional nonlocal phase-field and hyperbolic nonlocal phase-field models to obtain their numerical solutions. For this purpose, we propose a localized method based on radial basis functions (RBFs), namely localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for spatial discretization. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. This approach does not require meshing in spatial domain and hence inherits the meshless and spectral convergence properties of the global radial basis functions collocation method (GRBFCM). Some numerical results indicate that the obtained simulations via the LRBF-PSM is effective and stable for approximating the solution of nonlocal models investigated in the current paper.