Two-step discretization method for 2D/3D Allen-Cahn equation base on
RBF-FD scheme
Abstract
Based on the multi-quadrics (MQ) radial basis function, the two-step
discretization method will be used to solve Allen-Cahn (AC) equation in
this paper. In the first step, we will use the Backward Euler Formula to
solve the AC equation with Newton iterative, and then apply the global
RBF method to solve the Semi discrete linearized problem on a small
number of collocation points or on a coarse mesh. In the second step,
finite difference (FD) and radial basis function-finite difference
(RBF-FD) methods will be used to solve the problem on a fine mesh or on
a large number of collocation points, respectively. The two-step method
will be proved to be more feasible and efficient by some numerical
examples. In addition, compared with FD method, the application of
RBF-FD method will improve the accuracy of the numerical solution in the
second step.