A maximum bound principle preserving iteration technique for a class of semilinear parabolic equations

Abstract

This paper presents a systematic methodology, called the MBP-preserving iteration technique, to develop the maximum bound principle (MBP) preserving numerical algorithms for a class of semilinear parabolic equations. Two types of MBP-preserving iterations are suggested to solve two well-known θ-weighted schemes, respectively. Within some mild time-step constraints, the corresponding iteration solutions are proved to preserve the MBP property at each iteration step so that the numerical scheme has a uniquely MBP-preserving solution. In addition, concise error estimates in the maximum norm are established on nonuniform time meshes. Several numerical examples coupled with an adaptive time-stepping strategy are implemented for the Allen-Cahn model to confirm the theoretical findings and demonstrate their effectiveness for long-time numerical simulations.