Geodesic
Discrete maximum-norm stability of a~linearized second order finite difference scheme for Allen--Cahn equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 2, pp. 215-222

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In this paper, we use finite difference methods for solving the Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three level scheme in time and a second-order finite difference approach in space, and we establish discrete boundedness stability in maximum norm: if the initial data is bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. We will show that the main result can be obtained under certain. 
@article{SJVM_2017_20_2_a7,
     author = {T. Hou and K. Wang and Y. Xiong and X. Xiao and Sh. Zhang},
     title = {Discrete maximum-norm stability of a~linearized second order finite difference scheme for {Allen--Cahn} equation},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {215--222},
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     url = {http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a7/}
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T. Hou; K. Wang; Y. Xiong; X. Xiao; Sh. Zhang. Discrete maximum-norm stability of a~linearized second order finite difference scheme for Allen--Cahn equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 2, pp. 215-222. http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a7/