Error analysis of stabilized semi-implicit method of Allen-Cahn equation

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June 2009, 11(4): 1057-1070. doi: 10.3934/dcdsb.2009.11.1057

Error analysis of stabilized semi-implicit method of Allen-Cahn equation

1.	Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States

Received July 2008 Revised October 2008 Published April 2009

Abstract
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We consider in this paper the stabilized semi-implicit (in time) scheme and the splitting scheme for the Allen-Cahn equation $\phi_t-\Delta\phi+$ε$^-2f(\phi)=0$ arising from phase transitions in material science. For the stabilized first-order scheme, we show that it is unconditionally stable and the error bound depends on ε^-1 in some lower polynomial order using the spectrum estimate of [2, 10, 11]. In addition, the first- and second-order operator splitting schemes are proposed and the accuracy are tested and compared with the semi-implicit schemes numerically.

Keywords: Allen-Cahn equation, error analysis., semi-implicit methods.

Mathematics Subject Classification: Primary: 65M12, 76T99; Secondary: 65M7.

Citation: Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057

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