Numerical study of Allen–Cahn equation with variable diffusive coefficients

Numerical solutions of Allen–Cahn equation with variable diffusive coefficient are studied. First, we give some analysis to the steady-state Allen–Cahn equation, where by using the shooting method, we find three classical solutions to this nonlinear boundary value problem. This verifies the existence of steady state of time-dependent Allen–Cahn equation when time variable approaches to infinity. A standard fully discrete scheme for new Allen–Cahn equation is considered, where the second-order stabilized CNAB approximation in time and a central difference discretization in space are employed. Further, bearing a resemblance to numerical analysis for Allen–Cahn equation with constant coefficients, we prove the numerical scheme satisfies the discrete maximum principle under some mild constraints. Based on the numerical stability, the nonlinear energy stability of the fully discrete scheme is established, and the corresponding error estimate is established. Finally, numerical experiments with particular space-dependent coefficients are carried out which support the theoretical results of our investigation.