Stability analysis and error estimation based on difference spectral approximation for Allen–Cahn equation in a circular domain

Abstract

For the first time, we propose an efficient difference spectral approximation for Allen–Cahn equation in a circular domain. Firstly, we introduce the polar coordinate transformation and derive the equivalent form of Allen–Cahn equation under this coordinate system, as well as the corresponding essential polar condition. Then, by using first‐order Euler and second‐order backward difference methods in the temporal direction, we deduce the first‐order and second‐order semi‐implicit schemes, based on which the first‐order and second‐order fully discrete schemes are established by employing Legendre‐Fourier spectral approximation in the spatial direction. In addition, the energy stability and error estimations for the two types of numerical schemes are theoretically proved. Finally, we provide some numerical examples, the results of which demonstrate the stability and convergence of the algorithm.