A class of singularly perturbed quasilinear differential equations with interior layers

Farrell, P.; O’Riordan, E.; Shishkin, G.

doi:10.1090/S0025-5718-08-02157-1

A class of singularly perturbed quasilinear differential equations with interior layers
HTML articles powered by AMS MathViewer

by P. A. Farrell, E. O’Riordan and G. I. Shishkin;

Math. Comp. 78 (2009), 103-127

DOI: https://doi.org/10.1090/S0025-5718-08-02157-1

Published electronically: June 27, 2008

PDF | Request permission

Abstract:

A class of singularly perturbed quasilinear differential equations with discontinuous data is examined. In general, interior layers will appear in the solutions of problems from this class. A numerical method is constructed for this problem class, which involves an appropriate piecewise-uniform mesh. The method is shown to be a parameter-uniform numerical method with respect to the singular perturbation parameter. Numerical results are presented, which support the theoretical results.

References

Stephen R. Bernfeld and V. Lakshmikantham, An introduction to nonlinear boundary value problems, Mathematics in Science and Engineering, Vol. 109, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 445048
K. W. Chang and F. A. Howes, Nonlinear singular perturbation phenomena: theory and applications, Applied Mathematical Sciences, vol. 56, Springer-Verlag, New York, 1984. MR 764395, DOI 10.1007/978-1-4612-1114-3
P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient, Math. Comput. Modelling 40 (2004), no. 11-12, 1375–1392. MR 2127720, DOI 10.1016/j.mcm.2005.01.025
P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton), vol. 16, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1750671
Paul A. Farrell, John J. H. Miller, Eugene O’Riordan, and Grigorii I. Shishkin, On the non-existence of $\epsilon$-uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems, Math. Comp. 67 (1998), no. 222, 603–617. MR 1451321, DOI 10.1090/S0025-5718-98-00922-3
Paul A. Farrell, John J. H. Miller, Eugene O’Riordan, and Grigori I. Shishkin, A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1135–1149. MR 1393906, DOI 10.1137/0733056
Paul A. Farrell, Eugene O’Riordan, John J. H. Miller, and Grigorii I. Shishkin, Parameter-uniform fitted mesh method for quasilinear differential equations with boundary layers, Comput. Methods Appl. Math. 1 (2001), no. 2, 154–172. MR 1854309, DOI 10.2478/cmam-2001-0011
P. A. Farrell, E. O’Riordan, and G. I. Shishkin, A class of singularly perturbed semilinear differential equations with interior layers, Math. Comp. 74 (2005), no. 252, 1759–1776. MR 2164095, DOI 10.1090/S0025-5718-05-01764-3
Natalia Kopteva and Torsten Linß, Uniform second-order pointwise convergence of a central difference approximation for a quasilinear convection-diffusion problem, J. Comput. Appl. Math. 137 (2001), no. 2, 257–267. MR 1865231, DOI 10.1016/S0377-0427(01)00353-3
Torsten Linss, Hans-Görg Roos, and Relja Vulanović, Uniform pointwise convergence on Shishkin-type meshes for quasi-linear convection-diffusion problems, SIAM J. Numer. Anal. 38 (2000), no. 3, 897–912. MR 1781208, DOI 10.1137/S0036142999355957
J. Lorenz, Nonlinear singular perturbation problems and the Engquist-Osher difference scheme, Report 8115, University of Nijmegen , 1981.
J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Error estimates in the maximum norm for linear problems in one and two dimensions. MR 1439750, DOI 10.1142/2933
H.-G. Roos, M. Stynes, and L. Tobiska, Numerical methods for singularly perturbed differential equations, Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems. MR 1477665, DOI 10.1007/978-3-662-03206-0
G. I. Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic equations, Russian Academy of Sciences, Ural section, Ekaterinburg, (1992). (in Russian)
Relja Vulanović, A priori meshes for singularly perturbed quasilinear two-point boundary value problems, IMA J. Numer. Anal. 21 (2001), no. 1, 349–366. MR 1812279, DOI 10.1093/imanum/21.1.349