A simple and practical finite difference method for the phase-field crystal model with a strong nonlinear vacancy potential on 3D surfaces

The crystallization is a typical process in material science and this process can be described by the phase-field crystal type models. To investigate the dynamics of phase-field crystal model with a nonlinear vacancy potential on 3D surfaces, we herein propose a simple and practical finite difference method. The surfaces are defined by the zero level-set of signed distance functions and are included in a fully three-dimensional domain. By defining appropriate pseudo-Neumann boundary condition and using closest-point type method, the computation on surfaces are transformed into a three-dimensional narrow band domain containing the surfaces. Therefore, the finite difference method can be adopted to perform the spatial discretization. A linear semi-implicit discretization with stabilization technique in time is considered. In each time step, we only need to solve the elliptic type equations with constant coefficients. The numerical implementation is highly efficient. The numerical results indicate that the proposed method not only has desired accuracy but also works well for the pattern formations on various curved surfaces.