article

An adaptive splitting approach for the quenching solution of reaction-diffusion equations over nonuniform grids

Authors:

Matthew A. Beauregard,

Qin ShengAuthors Info & Claims

Journal of Computational and Applied Mathematics, Volume 241

Pages 30 - 44

https://doi.org/10.1016/j.cam.2012.10.005

Published: 01 March 2013 Publication History

Abstract

The numerical solution of a nonlinear degenerate reaction-diffusion equation of the quenching type is investigated. While spatial derivatives are discretized over symmetric nonuniform meshes, a Peaceman-Rachford splitting method is employed to advance solutions of the semidiscretized system. The temporal step is determined adaptively through a suitable arc-length monitor function. A criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution. Weak stability is proven in a von Neumann sense via the ~-norm. Computational examples are presented to illustrate our results.

References

[1]

Acker, A. and Kawohl, B., Remarks on quenching. J. Nonlinear Anal. v13 i1. 53-61.

[2]

Boni, T., Extinction for discretizations of some semilinear parabolic equations. Acad. Sci. Paris. Sér. I Math. v8. 795-800.

[3]

Nouaili, N., A Liouville theorem for a heat equation and applications for quenching. Nonlinearity. v24. 797-832.

[4]

Sheng, Q. and Khaliq, A., A compound adaptive approach to degenerate nonlinear quenching problems. Numer. Methods Partial Differential Equations. v15. 29-47.

[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. v25. 1273-1292.

Digital Library

[6]

Kirk, C. and Olmstead, W., Blow-up in a reactive-diffusive medium with a moving heat source. J. Zeitschrift Angew. Math. Phys. v53. 147-159.

Digital Library

[7]

Sheng, Q. and Khaliq, A., Adaptive Method of Lines. 2001. CRC Press, London, New York.

[8]

Ferreira, P., Numerical quenching for the semilinear heat equation with a singular absorption. J. Comput. Appl. Math. v228. 92-103.

Digital Library

[9]

Levine, H., Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations. Ann. Math. Pure Appl. v4. 243-260.

[10]

Liang, K., Lin, P., Ong, M. and Tan, R., A splitting moving mesh method for reaction-diffusion equations of quenching type. J. Comput. Phys. v215. 757-777.

Digital Library

[11]

Mooney, J., A numerical method for accurate critical length estimation in singular quenching problems. World Sci. Ser. Appl. Anal. v4. 505-516.

[12]

N'Gohisse, F. and Boni, T., Quenching time of some nonlinear wave equations. Arch. Math. v45. 115-124.

[13]

Wang, W. and Zheng, S., Asymptotic estimates to quenching solutions of heat equations with weighted absorptions. Asymp. Anal. v70. 125-139.

[14]

Bebernes, J. and Eberly, D., Mathematical Problems from Combustion Theory, Vol. 83. 1989. Springer-Verlag.

[15]

Floater, M., Blow-up at the boundary for degenerate semilinear parabolic equations. Arch. Ration. Mech. Anal. v114. 57-77.

[16]

Ockendon, H., Channel flow with temperature-dependent viscosity and internal viscous dissipation. J. Fluid Mech. v93. 737-746.

[17]

Boni, T., On quenching of solution for some semilinear parabolic equation of second order. Bull. Belg. Math. Soc. v5. 73-95.

[18]

Chan, C. and Ke, L., Parabolic quenching for nonsmooth convex domains. J. Math. Anal. Appl. v186. 52-65.

[19]

Furzeland, R., Verwer, J. and Zegeling, P., 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. v89. 349-388.

Digital Library

[20]

Lang, J. and Walter, A., An adaptive rothe method for nonlinear reaction-diffusion systems. Appl. Numer. Math. v13. 135-146.

Digital Library

[21]

Sheng, Q., Adaptive decomposition finite difference methods for solving singular problems. Frontiers Math. China. v4. 599-626.

[22]

An adaptive grid method for degenerate seminlinear quenching problems. Compters Math. Appl. v39. 57-71.

[23]

A fundamental theorem on the structure of symplectic integrators. Phys. Lett. A. v354. 373-376.

[24]

Jain, B. and Sheng, A., An exploration of the approximation of derivative functions via finite differences. Rose-Hulman Undergrd. Math J. v8. 172-188.

[25]

Henrici, P., Discrete Variable Methods in Ordinary Differential Equations. 1962. John Wiley & Sons, Inc., New York.

[26]

Varah, J., A lower bound for the smallest singular value of a matrix. Linear Algebra Appl. v11. 3-5.

Cited By

Beauregard MPadgett JParshad R(2017)A nonlinear splitting algorithm for systems of partial differential equations with self-diffusionJournal of Computational and Applied Mathematics10.1016/j.cam.2017.02.019321:C(8-25)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1016/j.cam.2017.02.019
Chung SPark J(2016)A complete characterization of nonlinear absorption for the evolution p -Laplacian equations to have positive or extinctive solutionsComputers & Mathematics with Applications10.1016/j.camwa.2016.02.02171:8(1624-1635)Online publication date: 1-Apr-2016
https://dl.acm.org/doi/10.1016/j.camwa.2016.02.021
Beauregard M(2015)Numerical solutions to singular reaction-diffusion equation over elliptical domainsApplied Mathematics and Computation10.1016/j.amc.2014.12.086254:C(75-91)Online publication date: 1-Mar-2015
https://dl.acm.org/doi/10.1016/j.amc.2014.12.086
Show More Cited By

An adaptive splitting approach for the quenching solution of reaction-diffusion equations over nonuniform grids
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
2. Theory of computation

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cover image Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Volume 241, Issue

March, 2013

142 pages

ISSN:0377-0427

Issue’s Table of Contents

Copyright © Elsevier B.V. © 2012.

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Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 March 2013

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Cited By

Beauregard MPadgett JParshad R(2017)A nonlinear splitting algorithm for systems of partial differential equations with self-diffusionJournal of Computational and Applied Mathematics10.1016/j.cam.2017.02.019321:C(8-25)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1016/j.cam.2017.02.019
Chung SPark J(2016)A complete characterization of nonlinear absorption for the evolution p -Laplacian equations to have positive or extinctive solutionsComputers & Mathematics with Applications10.1016/j.camwa.2016.02.02171:8(1624-1635)Online publication date: 1-Apr-2016
https://dl.acm.org/doi/10.1016/j.camwa.2016.02.021
Beauregard M(2015)Numerical solutions to singular reaction-diffusion equation over elliptical domainsApplied Mathematics and Computation10.1016/j.amc.2014.12.086254:C(75-91)Online publication date: 1-Mar-2015
https://dl.acm.org/doi/10.1016/j.amc.2014.12.086
Sheng QSun H(2014)Exponential splitting for n-dimensional paraxial Helmholtz equation with high wavenumbersComputers & Mathematics with Applications10.1016/j.camwa.2014.09.00568:10(1341-1354)Online publication date: 1-Nov-2014
https://dl.acm.org/doi/10.1016/j.camwa.2014.09.005

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