Communications in Mathematical Sciences

Volume 20 (2022)

Number 5

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

Pages: 1409 – 1436



Jie Shen (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Xiangxiong Zhang (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


We consider solving a generalized Allen–Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to the construction of a bound-preserving scheme for any passive convection with an incompressible velocity field.


discrete maximum principle, high order accuracy, monotonicity, bound-preserving, phase field equations, incompressible flow

2010 Mathematics Subject Classification

65M06, 65M12, 65M60

1fundingJ. Shen is supported in part by NSF grant DMS-2012585 and AFOSR grant FA9550-20-1-0309, while X. Zhang is supported in part by the NSF grant DMS-1913120.

Received 22 April 2021

Received revised 1 December 2021

Accepted 23 December 2021

Published 26 May 2022