Abstract: In this paper, a semi-implicit collocation scheme and the corresponding discrete linear equation system are firstly derived for the Allen-Cahn equation by discretizing it in spatial direction with the barycentric Lagrange interpolation collocation method, in time direction with the backward Euler scheme and the Crank-Nicolson scheme respectively and treating its nonlinear term with an explicit scheme. Then, the consistency of the one-dimensional spatial semi-discrete schemes and the two-dimensional full discrete schemes are analyzed. Finally, the high accuracy and energy decreasing law of the semi-implicit collocation scheme are verified by numerical examples. Comparing with the two kinds of classical difference schemes, the numerical results show that the proposed scheme can achieve higher accuracy with fewer nodes.

Key words: Allen-Cahn equation, Barycentric interpolation collocation method, Semi-implicit scheme , Consistency analysis, Energy decline