Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T04:10:22.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

Published online by Cambridge University Press:  03 June 2015

Matthew A. Beauregard  and
Qin Sheng
Matthew A. Beauregard*
Affiliation:
Department of Mathematics, Baylor University, TX 76798-7328, USA
Qin Sheng*
Affiliation:
Department of Mathematics, Baylor University, TX 76798-7328, USA
*
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

Abstract

Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-called primitive regime, the rest belong to a later category of the modified type, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

Keywords

65K2065M5035K65Degeneracyquenching singularityadaptive difference methodarc-lengthmonitoring functionsplitting method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 407 - 422
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Beauregard, M. A. and Sheng, Q., A semi-adaptive compact splitting method for the numerical solution of 2-dimensional quenching problems, Appl. Math. Comput., 218 (2012), pp. 1240–1254.Google Scholar
[2]Beauregard, M. A. and Sheng, Q., Solving degenerate quenching-combustion equations by an adaptive splitting method on evolving grids, Comput. Struct., 122 (2013), pp. 33–43.Google Scholar
[3]Budd, C., Huang, W. and Russell, R., Adaptivity with moving grids, Acta Numer., (2009), pp. 111–241.CrossRefGoogle Scholar
[4]Chan, C., A quenching criterion for a multi-dimensional parabolic problem due to a concentrated nonlinear source, J. Comput. Appl. Math., 235 (2011), pp. 3724–3727.Google Scholar
[5]Cheng, H., Lin, P., Sheng, Q. and Tan, R., Solving degenerate reaction-diffusion equations via variable step peaceman-rachford splitting, SIAM J. Sci. Comput., 25(4) (2003), pp. 1273–1292.Google Scholar
[6]Ferreira, P., Numerical quenching for the semilinear heat equation with a singular absorption, J. Comput. Appl. Math., 228 (2009), pp. 92–103.Google Scholar
[7]Kawarada, H., On solutions of initial-boundary problem for , Publ. Res. Inst. Math. Sci., 10 (1975), pp. 729–736.CrossRefGoogle Scholar
[8]Khaliq, A. and Sheng, Q., On the monotonicity of an adaptive splitting scheme for two-dimensional singular reactiondiffusion equations, Int. J. Comput. Math., 84 (2007), pp.795–806.Google Scholar
[9]Kirk, C. and Olmstead, W., Blow-up in a reactive-diffusive medium with a moving heat source, J. Zeitschrift Angew. Math. Phys., 53 (2002), pp. 147–159.Google Scholar
[10]Levine, H., Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations, Ann. Math. Pure Appl., 4 (1989), pp. 243–260.Google Scholar
[11]Liang, K., Lin, P., Ong, M. and Tan, R., A splitting moving mesh method for reaction-diffusion equations of quenching type, J. Comput. Phys., 215 (2006), pp. 757–777.CrossRefGoogle Scholar
[12]Liang, K., Lin, P. and Tan, R., Numerical solution of quenching problems using mesh-dependent variable temporal steps, Appl. Numer. Math., 57 (2007), pp. 791–800.Google Scholar
[13]N’GOhisse, F. and Boni, T., Quenching time of some nonlinear wave equations, Arch. Mathematicum, 45 (2009), pp. 115–124.Google Scholar
[14]Nouaili, N., A liouville theorem for a heat equation and applications for quenching, Nonlinearity, 24 (2011), pp. 797–832.Google Scholar
[15]Sheng, Q., Adaptive decomposition finite difference methods for solving singular problems, Frontiers Math. China, 4 (2009), pp. 599–626.Google Scholar
[16]Sheng, Q. and Cheng, H., An adaptive grid method for degenerate semilinear quenching problems, Compt. Math. Appl., 39 (2000), pp. 57–71.CrossRefGoogle Scholar
[17]Sheng, Q. and Khaliq, A., Adaptive Method of Lines, Chapter 9, CRC Press, London and New York, 2001.Google Scholar