A meshless technique based on generalized moving least squares combined with the second-order semi-implicit backward differential formula for numerically solving time-dependent phase field models on the spheres

Abstract

In the current research paper, the generalized moving least squares technique is considered to approximate the spatial variables of two time-dependent phase field partial differential equations on the spheres in Cartesian coordinate. This is known as a direct approximation (it is the standard technique for generalized finite difference scheme [69,77]), and it can be applied for scattered points on each local sub-domain. The main advantage of this approach is to approximate the Laplace-Beltrami operator on the spheres using different types of distribution points simply, in which the studied mathematical models are involved. In fact, this scheme permits us to solve a given partial differential equation on the sphere directly without changing the original problem to a problem on a narrow band domain with pseudo–Neumann boundary conditions. A second-order semi-implicit backward differential formula (by adding a stabilized term to the chemical potential that is the second-order Douglas-Dupont-type regularization) is applied to approximate the temporal variable. We show that the time discretization considered here guarantees the mass conservation and energy stability. Besides, the convergence analysis of the proposed time discretization is given. The resulting fully discrete scheme of each partial differential equation is a linear system of algebraic equations per time step that is solved via an iterative method, namely biconjugate gradient stabilized algorithm. Some numerical experiments are presented to simulate the phase field Cahn-Hilliard, nonlocal Cahn-Hilliard (for diblock copolymers as microphase separation patterns) and crystal equations on the two-dimensional spheres.