Temporal high-order, unconditionally maximum-principle-preserving integrating factor multi-step methods for Allen-Cahn-type parabolic equations

We develop a class of explicit unconditionally maximum-principle-preserving multi-step methods for the Allen-Cahn-type semilinear parabolic equations. Based on the usual second-order finite difference space discretization, we first show that the strong-stability-preserving (SSP) integrating factor multi-step method can preserve the maximum-principle under the condition τ≤CτFE, where C is the SSP coefficient and τFE is independent of the space step size, thus avoiding the strict parabolic Courant-Friedrichs-Lewy condition τ=O(h2). By introducing the stabilization technique and replacing the exponential functions with two different approximations, we obtain two types of improved stabilized integrating factor multi-step (isIFMS) schemes. The unconditional maximum-principle-preservation, mass-conservation, linear stability, and error estimates are analyzed for the isIFMS schemes. Since explicit SSP multi-step methods have no order barrier, the proposed framework offers a concise but effective approach for developing temporal high-order, unconditionally maximum-principle-preserving schemes for Allen-Cahn-type equations, and mass-conserving schemes for their conservative reformulations. Finally, numerical experiments validate theoretical results of the proposed isIFMS schemes.