A reduced‐order modified finite difference method preserving unconditional energy‐stability for the Allen–Cahn equation

Huanrong Li, Zhengyuan Song, Fuchen Zhang

2020Numerical Methods for Partial Differential Equations27 citationsDOI

Abstract

Abstract In this paper, we mainly study a reduced‐order finite difference (FD) method with an extra modified term for the Allen–Cahn equation with a small parameter perturbation and a nonlinear term concerned with the energy function. First, using a proper orthogonal decomposition (POD) technique, we construct a set of optimal POD basis and establish a reduced‐order modified finite difference (ROMFD) scheme. Second, we prove the discrete maximum‐bound‐principle (DMBP) preserving and discrete energy‐stability (DES) preserving of the ROMFD solutions under some restrictions on the coefficient of the modified term and the time step size, in particular when the coefficient of modified term is large enough, the ROMFD scheme satisfies unconditional DMP and unconditional DES for any time step. And we analyze the error estimate of the ROMFD solution for the Allen–Cahn equation. Finally, we give some numerical tests to verify all the theoretical results of our ROMFD method, including (unconditional) DMBP‐preserving and (unconditional) DES‐preserving, and to show that the CPU running time of our ROMFD method with very few degrees of freedom is much less than that of the FD method.

Topics & Concepts

MathematicsTerm (time)Applied mathematicsStability (learning theory)Finite differenceFunction (biology)Allen–Cahn equationMathematical analysisPhysicsBiologyComputer scienceQuantum mechanicsMachine learningEvolutionary biologyNumerical methods for differential equationsAdvanced Numerical Methods in Computational MathematicsDifferential Equations and Numerical Methods