Solving Allen-Cahn equations with periodic and nonperiodic boundary conditions using mimetic finite-difference operators

Abstract

[EN] In this paper, we investigate and implement a numerical method that is based on the mimetic finite difference operator in order to solve the nonlinear Allen-Cahn equation with periodic and non-periodic boundary conditions. In addition, we also analyze the performance of this mimetic- based method by using the classical heat equation with a variety of boundary conditions. We assess the performance of the mimetic-based numerical method by comparing the errors of its solutions with those obtained by a classical finite difference method and the pdepde built-in Matlab function. We compute the errors by using the exact solutions when they are available or with reference solutions. We adapt and implement the mimetic-based numerical method by using the MOLE (Mimetic Operators Library Enhanced) library that includes some built-in functions that return representations of the curl, divergence and gradient operators, in order to deal with the Allen-Cahn and heat equations. We present several results with regard to errors and numerical convergence tests in order to provide insight into the accuracy of the mimetic-based numerical method. The results show that the numerical method based on the mimetic difference operator is a reliable method for solving the Allen-Cahn and heat equations with periodic and non- periodic boundary conditions. The numerical solutions generated by the mimetic-based method are relatively accurate. We also proposed a new method based on the mimetic finite difference operator and the convexity splitting approach to solve Allen-Cahn equation in 2D. We found that, for small time step sizes the solutions generated by the mimetic-based method are more accurate than the ones generated by the pdepe Matlab function and similar to the solutions given by a finite difference method.S.O is grateful for the start-up funds provided by New Mexico Tech. S.O thanks Thomas Wilteski (Duke University) for hosting a research visit during summer 2023. G.G.P thanks Universitat Politecnica de Valencia for a year long visit under grant Maria Zambrano (UPV, funding from the Spain Ministry of Universities funded by the European Union-Next Generation EU) . Research reported in this publication was supported by an Institutional Development Award (IDeA) from the National Institute of General Medical Sciences of the National Institutes of Health under grant number P20GM103451.Orizaga, S.; González Parra, GC.; Forman, L.; Villegas-Villanueva, J. (2025). Solving Allen-Cahn equations with periodic and nonperiodic boundary conditions using mimetic finite-difference operators. Applied Mathematics and Computation. 484. https://doi.org/10.1016/j.amc.2024.12899348