Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle

Abstract

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

“…To solve the Allen-Cahn equations numerically, it is known that the simple semi-and fully implicit schemes requires the condition that t ≤ C 2 to guarantee gradient stability (energy decay), maximum principle preservation and even existence of discrete solution [11][12][13], where t is the time step size and C is a constant. To solve the Allen-Cahn equations numerically, it is known that the simple semi-and fully implicit schemes requires the condition that t ≤ C 2 to guarantee gradient stability (energy decay), maximum principle preservation and even existence of discrete solution [11][12][13], where t is the time step size and C is a constant.…”

“…In terms of this topic, the stabilized semi-implicit finite difference scheme is developed and has been proved to be maximum principle preserving [13,14,21]. However, the stronger stability under the infinity norm has a great effect on avoiding numerical oscillations.…”

Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

“…Numerical computations are essentially important to understand the behavior of the solution. In the existing literature, the Allen-Cahn equation was numerically extensively studied [9][10][11][12][13][14][15][16][17][18][19]. For example, the Allen-Cahn equation was numerically solved by the operator splitting scheme [17][18][19].…”

“…And Choi et al [9] proposed an unconditionally gradient stable nonlinear scheme with both discrete maximum principle and energy decreasing properties. The discrete maximum principle is also discussed in [15,16] for the Allen-Cahn equation and the generalized Allen-Cahn equation, respectively. And the discrete energy stability is thoroughly analysed by several scientists [10][11][12][13][14][15].…”

Stability and Convergence Analysis of Second-Order Schemes for a Diffuse Interface Model with Peng-Robinson Equation of State

“…Numerical experiments are our indispensable tools to investigate the solution to this equation. Previously proposed schemes for the CahnHilliard equation [7,8,[10][11][12][13][14]17,29,34,35,38] and other kinetics equations contain fourth order term [23,25,26,28,37] could be used as valuable references. The main contribution of this work is to develop two second-order energy stable numeircal schemes for the two-dimensional diffuse interface model with Peng-Robinson EOS of single component substance.…”