Approximate solutions to the Allen–Cahn equation using rational radial basis functions method

We apply the rational radial basis functions (RRBFs) method to solve the Allen–Cahn (A.C) equation, particularly when the equation has a so-lution with steep front or sharp gradients. We approximate the spatial derivatives by the RRBFs method. Then we apply an explicit, fourth-order Runge–Kutta method to advance the resulting semi-discrete system in time. It is well known that the A.C equation has a nonlinear stability feature, meaning that the free-energy functional is reduced by time. The presented method maintains the total energy reduction property of the A.C equation. In the end, five examples to confirm the efficiency and accuracyof the proposed method are provided.

Keywords:

Allen–Cahn equation , RBFs , Rational RBFs method , Runge–Kutta method