Abstract
Numerical methods are considered for singularly perturbed quasilinear problems having interior-shock solutions. It is shown that the direct discretization on a layer-adapted mesh is ineffective for these problems. A special method is proposed for the case when the solution is monotonic: the problem is transformed by interchanging the dependent and independent variables, and it is then discretized on a uniform mesh. The method is analyzed both theoretically and numerically. It is shown that it can be effective, but that it is not entirely without problems. An approach for improving the method is suggested.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allgower, E.L., Mccormickm, St.F., Pryor, D.V.: A general mesh independence principle for Newton’s method applied to second order boundary value problems. Computing 23, 233–246 (1979)
Bohl, E.: Finite Modelle Gewöhnlicher Randwertaufgaben. Teubner, Stuttgart (1981)
Chang, K.W., Howes, F.A.: Nonlinear Singular Perturbation Phenomena: Theory and Application (Applied Mathematical Sciences, vol. 56. Springer, New York (1984)
Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC, Boca Raton (2000)
Farrell, P.A., O’Riordan, E., Shishkin, G.I.: A class of singularly perturbed quasilinear differential equations with interior layers. Math. Comput. 78, 103–127 (2009)
Herman, R.L., Knickerbocker, C.J.: Numerically induced phase shift in the KdV soliton. J. Comput. Phys. 104, 50–55 (1993)
Howes, F.A.: Boundary-Interior Layer interactions in nonlinear singular perturbation theory. Mem. Amer. Math. Soc. 203 (1978)
Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics (Applied Mathematical Sciences, vol. 34). Springer, New York (1980)
Lagerstrom, P.A.: Matched Asymptotic Expansions (Applied Mathematical Sciences, vol. 76). Springer, New York (1988)
Linß, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems (Lecture Notes in Mathematics, vol. 1985). Springer, Berlin, Heidelberg (2010)
Lorenz, J.: Stability and monotonicity properties of stiff quasilinear boundary problems. Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 12, 151–175 (1982)
Lorenz, J.: Analysis of difference schemes for a stationary shock problem. SIAM J. Numer. Anal. 21, 1038–1053 (1984)
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singularly Perturbation Problems. World Scientific, Singapore (1996)
O’Malley, R.E. Jr: Singular Perturbation Methods for Ordinary Differential Equations (Applied Mathematical Sciences, vol. 89). Springer, New York (1991)
O’Riordan, E., Quinn, J.: Numerical Method for a Nonlinear Singularly Perturbed Interior Layer Problem. In: Clavero, C., Gracia, J.L., Lisbona, F. (eds.) Proceedings BAIL 2010 (Lecture Notes in Computational Science and Engineering 81), pp. 187–195. Springer, Berlin, Heidelberg (2011)
O’Riordan, E., Quinn, J.: Parameter-uniform numerical methods for some linear and nonlinear singularly perturbed convection diffusion boundary turning point problems. BIT 51, 317–337 (2011)
O’Riordan, E., Quinn, J.: A singularly perturbed convection diffusion turning point problem with an interior layer. J. Comp. Meth. Appl. Math. 12, 206–220 (2012)
O’Riordan, E., Quinn, J.: Numerical experiments with a Shishkin numerical method for a singularly perturbed quasilinear parabolic problem with an interior layer. In: Dimov, I., Farago, I., Vulkov, L. (eds.) Numerical Analysis and Its Applications: 5th International Conference, NAA 2012, Lozenetz, Bulgaria, June 15–20, 2012, Revised Selected Papers, (Lecture Notes in Computer Science 8236), pp. 420–427. Springer, Heidelberg (2013)
Osher, S.: Nonlinear singular perturbation problems and one sided difference schemes. SIAM J. Numer. Anal. 18, 129–144 (1981)
Roos, H.-G., Stynes, M., Tobiska L.: Numerical Methods for Singularly Perturbed Differential Equations (Springer Series in Computational Mathematics, 2nd edn, vol. 24). Springer, Berlin (2008)
Shishkin, G.I.: Grid approximation of a singularly perturbed quasilinear equation in the presence of a transition layer. Russian Acad. Sci. Dokl. Math. 47, 83–88 (1993)
Shishkin, G.I.: Difference approximation of the Dirichlet problem for a singularly perturbed quasilinear parabolic equation in the presence of a transition layer. Russian Acad. Sci. Dokl. Math. 48, 346–352 (1994)
Vulanović, R.: Finite-difference schemes for quasilinear singular perturbation problems. J. Comp. Appl. Math. 26, 345–365 (1989)
Vulanović, R.: Continuous and numerical analysis of a boundary shock problem. Bull. Austral. Math. Soc. 41, 75–86 (1990)
Vulanović, R.: A uniform numerical method for a class of quasilinear turning point problems. In: Miller, J.J.H., Vichnevetsky, R. (eds.) Proceedings 13th IMACS World Congress on Computation and Applied Mathematics, p. 493. IMACS, Dublin (1991)
Vulanović, R.: Numerical methods for quadratic singular perturbation problems. Proc. Dynamic Systems and Appl. 2, 543–551 (1996)
Vulanović, R.: Boundary shock problems and singularly perturbed Riccati equations. In: Hegarty, A.F., Kopteva, N., O’Riordan, E., Stynes, M. (eds.) Proceedings BAIL 2008 (Lecture Notes in Computational Science and Engineering 69), pp. 277–285. Springer, Berlin, Heidelberg (2009)
Zadorin, A.I.: Existence and uniqueness of the solution of certain difference problems for a quasilinear ordinary differential equation with a small parameter. (Russian) Chisl. Metody Mekh. Sploshn. Sredy 15, 33–44 (1984)
Acknowledgments
Thanks are due to two anonymous reviewers whose comments helped us improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vulanović, R., Nhan, T.A. A numerical method for stationary shock problems with monotonic solutions. Numer Algor 77, 1117–1139 (2018). https://doi.org/10.1007/s11075-017-0355-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-017-0355-4