Up to fourth-order unconditionally structure-preserving parametric single-step methods for semilinear parabolic equations

We propose and analyze a class of temporal up to fourth-order unconditionally structure-preserving single-step methods for Allen-Cahn-type semilinear parabolic equations. We first revisit some up to second-order exponential time different Runge-Kutta (ETDRK) schemes, and provide unified proofs for the unconditionally maximum-principle-preserving and mass-conserving properties. Noting that the stabilized ETDRK schemes belong to a special class of parametric Runge-Kutta schemes, we introduce the stabilized integrating factor Runge-Kutta (sIFRK) formulation to construct new high-order parametric single-step methods, and propose two strategies to eliminate the exponential effects of sIFRK: (1) a recursive approximation; (2) a combination of exponential and linear functions. Together with the nonnegativity of coefficients and non-decreasing of abscissas, the resulting two families of improved stabilized integrating factor Runge-Kutta (isIFRK) schemes can unconditionally preserve the maximum-principle and conserve the mass. The order conditions, linear stability and convergence in the l(infinity)- norm are analyzed rigorously. We demonstrate that the proposed framework, which is explicit and free of limiters or cut-off post-processing, offers a simple, practical, and effective approach to developing high-order unconditionally structure-preserving algorithms. Comparisons with traditional schemes demonstrate the necessity of developing high-order unconditionally structure-preserving schemes. A series of numerical experiments verify theoretical results of proposed isIFRK schemes. (C)& nbsp;2022 Elsevier B.V. All rights reserved.

Automated summary

In this study, a new high-order unconditionally structure-preserving single-step method is proposed for solving Allen-Cahn-type parabolic equations. The method can unconditionally preserve the maximum principle and conserve mass, and it does not require limiters or cutoff post-processing. It offers a simple, practical, and effective approach to developing high-order unconditionally structure-preserving algorithms.