Abstract: Compared with the well-known classical Allen--Cahn equation, the modified Allen--Cahn equation, equipped with a nonlocal Lagrange multiplier, enforces the mass conservation for modeling phase transitions. In this paper, a class of up to eighth-order maximum principle preserving schemes are proposed for solving the modified conservative Allen--Cahn equation. Based on the second-order finite-difference space discretization, we investigate the high-order integrating factor two-step Runge--Kutta maximum principle preserving schemes. We prove that the schemes can preserve the maximum principle and mass of the conservative Allen--Cahn equation and give the convergence analysis of proposed schemes. Finally, two- and three-dimensional numerical tests are carried out to verify the theoretical results and demonstrate the performance of proposed schemes.

Key words: Maximum principle preserving scheme, Modified Allen--Cahn equation, Mass conservation , Integrating factor two-step Runge--Kutta method