Explicit Third-Order Unconditionally Structure-Preserving Schemes for Conservative Allen-Cahn Equations

Compared with the well-known classical Allen-Cahn equation, the modified Allen-Cahn equation, which is equipped with a nonlocal Lagrange multiplier or a local-nonlocal Lagrange multiplier, enforces the mass conservation for modeling phase transitions. In this work, a class of up to third-order explicit structure-preserving schemes is proposed for solving these two modified conservative Allen-Cahn equations. Based on second-order finite-difference space discretization, we investigate the newly developed improved stabilized integrating factor Runge-Kutta (isIFRK) schemes for conservative Allen-Cahn equations. We prove that the original stabilized integrating factor Runge-Kutta schemes fail to preserve the mass conservation law when the stabilizing constant kappa > 0 and the initial mass does not equal zero, while isIFRK schemes not only preserve the maximum principle unconditionally, but also conserve the mass to machine accuracy without any restriction on the time-step size. Convergence of the proposed schemes are also presented. At last, a series of numerical experiments validate that each reformulation of the conservative Allen-Cahn equations has it own advantage, and isIFRK schemes can reach the expected high-order accuracy, conserve the mass, and preserve the maximum principle unconditionally.