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

Pages: 1057 - 1070, Volume 11, Issue 4, June 2009

doi:10.3934/dcdsb.2009.11.1057       Abstract        Full Text (569.8K)       Related Articles

Xiaofeng Yang - Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States (email)

Abstract: 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, semi-implicit methods, error analysis.
Mathematics Subject Classification:  Primary: 65M12, 76T99; Secondary: 65M70.

Received: July 2008;      Revised: October 2008;      Published: April 2009.