Linear Maximum Bound Principle Preserving Finite Difference Schemes for the Convective Allen-Cahn Equation

Abstract

The convective Allen-Cahn equation generalizes the classical Allen-Cahn equation by introducing an additional convective term associated with a solenoidal velocity field while maintaining the maximum bound principle (MBP). However, developing high-order numerical schemes that are accurate in both time and space and preserve the MBP unconditionally has remained a significant challenge. In this paper, we address this by first defining new auxiliary variables to reformulate the interaction of the velocity field with the phase field. We then transform the convective Allen-Cahn equation into a generalized Fokker-Planck form using an exponential transformation, enabling the development of MBP-preserving linear numerical schemes. Subsequently, we propose first- and second-order in time numerical schemes for the reformulated equations with a second-order quasi-symmetric finite difference discretization in space. In this approach, the auxiliary variables are replaced with known functions related to the velocity field, simplifying the numerical implementation. For the first-order in time scheme, we derive its optimal error estimate and prove its unconditional MBP-preservation. For the second-order in time scheme, we show its MBP-preservation under mild constraints on the mesh and time step sizes. Some numerical experiments in two and three dimensions are also presented to validate the theoretical findings and illustrate the accuracy and efficiency of our proposed schemes.