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A Revisit of the Semi-Adaptive Method for Singular Degenerate Reaction-Diffusion Equations

Published online by Cambridge University Press:  28 May 2015

Qin Sheng*
Affiliation:
Department of Mathematics, Center for Astrophysics, Space Physics and Engineering Research, Baylor University, Waco, TX 76798-7328, USA
A. Q. M. Khaliq
Affiliation:
Department of Mathematical Sciences, Center for Computational Science, Middle Tennessee State University, Murfreesboro, TN 37132, USA
*
Corresponding author. Email: Qin_Sheng@baylor.edu
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Abstract

This article discusses key characteristics of a semi-adaptive finite difference method for solving singular degenerate reaction-diffusion equations. Numerical stability, monotonicity, and convergence are investigated. Numerical experiments illustrate the discussion. The study reconfirms and improves several of our earlier results.

Type
Research Article
Copyright
Copyright © Global-Science Press 2012

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References

[1]Baker, G. A. and Graves-Morris, A., Padé Approximants, Cambridge University Press, London and New York, 1995.Google Scholar
[2]Bao, G., Wei, G. W. and Zhao, S., Numerical solution of the Helmholtz equation with high wavenumbers, Int. J. Numer. Meth. Engng., 59 (2004), 389408.Google Scholar
[3]Beauregard, M. and Sheng, Q., Solving degenerate quenching-combustion equations by an adaptive splitting method on evolving grids, Computers Struct., (2012), in press.Google Scholar
[4]Chan, C.Y. and Ke, L., Parabolic quenching for nonsmooth convex domains, J. Math. Anal. Appl., 186 (1994), 5265.CrossRefGoogle Scholar
[5]Cheng, H., Lin, P., Sheng, Q. and Tan, R. C. E., Solving degenerate reaction-diffusion equations via variable step Peaceman-Rachford splitting, SIAM J. Sci. Comput., 25 (2003), 12731292.CrossRefGoogle Scholar
[6]Ferreira, P., Numerical quenching for the semilinear heat equation with a singular absorption, J. Comput. Appl. Math., 228 (2009), 92103.CrossRefGoogle Scholar
[7]Furzeland, R. M., Verwer, J. G. and Zegeling, P. A., A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines, J. Comput. Phys., 89 (1990), 349388.Google Scholar
[8]Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2nd Ed., New York, 2011.Google Scholar
[9]Jain, B. and Sheng, A., An exploration of the approximation of derivative functions via finite differences, Rose-Hulman Undergrd. Math J., 8 (2007), 172188.Google Scholar
[10]Levine, H., Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations, Ann. Math. Pure Appl., 4 (1989), 243260.CrossRefGoogle Scholar
[11]Nouaili, N., A Liouville theorem for a heat equation and applications for quenching, Nonlinearity, 24 (2011), 797832.Google Scholar
[12]Poole, G. and Boullion, T., A survey on M-matrices, SIAM Review, 16 (1974), 419427.CrossRefGoogle Scholar
[13]Qiao, Z., Zhang, Z. and Tang, T., An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), 13951414.Google Scholar
[14]Sheng, Q., Adaptive decomposition finite difference methods for solving singular problems, Frontiers Math. China, 4 (2009), 599626.Google Scholar
[15]Sheng, Q. and Agarwal, R., Nonlinear variation of parameter methods for summary difference equations in several independent variables, J. Appl. Math. Comput., 61 (1994), 3960.CrossRefGoogle Scholar
[16]Sheng, Q. and Cheng, H., An adaptive grid method for degenerate semilinear quenching problems, Computers Math. Appl., 39 (2000), 5771.Google Scholar
[17]Sheng, Q. and Khaliq, A. Q. M., A compound adaptive approach to degenerate nonlinear quenching problems, Numer. Meth. Partial Diff. Eqns, 15 (1999), 2947.Google Scholar
[18]Sheng, Q. and Khaliq, A. Q. M., Linearly implicit adaptive schemes for singular reaction-diffusion equations, Chapter 9, Adaptive Method of Lines, CRC Press, London and New York, 2001.Google Scholar
[19]Sheng, Q. and Khaliq, A. Q. M., Modified arc-length adaptive algorithms for degenerate reaction-diffusion equations, Appl. Math. Comput., 126 (2002), 279297.Google Scholar
[20]Tan, Z. J., Zhang, Z. R., Huang, Y. Q. and Tang, T., Moving mesh methods with locally varying time steps, J. Comput. Phys., 200 (2004), 347367.CrossRefGoogle Scholar
[21]Trevelyan, J. and Honnor, M. E., A numerical coordinate transformation for efficient evaluation of oscillatory integrals over wave boundary elements, J. Integ. Eqn, 21 (2009), 447468.Google Scholar