The Spectral Collocation Method for the Stochastic Allen-Cahn Equation via Generalized Polynomial Chaos

Abstract

In this article, a spectral collocation algorithm is presented for the solution of the stochastic Allen-Cahn equation. Based on the generalized polynomial chaos method, the stochastic Allen-Cahn equation is turned into a set of nonlinear partial differential equations. Then the first-order fully implicit scheme is applied to solve the set of partial differential equations by using the Chebyshev spectral collocation method. Numerical results show that the maximum principle is satisfied. Moreover, numerical comparisons with the Monte Carlo method are reported which demonstrate the effectivity and feasibility of the proposed algorithm.