### IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE

Tao Tang1, Jiang Yang2

1. 1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China;
2. Department of Applied Mathematics, Columbia University, New York, NY 10027, USA
• Received:2014-08-18 Revised:2015-12-23 Online:2016-09-15 Published:2016-09-15

Tao Tang, Jiang Yang. IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE[J]. Journal of Computational Mathematics, 2016, 34(5): 451-461.

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.

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