Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-07T16:37:56.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous and numerical analysis of a boundary shock problem

Published online by Cambridge University Press:  17 April 2009

Relja Vulanović
Relja Vulanović
Affiliation:
Institute of Mathematics, University of Novi Sad, ul. Dr I Djuricica 4, Yu-21000 Novi Sad, Yugoslavia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A quasilinear singularly perturbed boundary value problem whose solution has a shock layer is investigated. Estimates of the derivatives of the solution are derived. Based on these estimates, a new independent variable is introduced. Then the transformed problem is solved numerically using finite - difference schemes. The transformation corresponds to solving the original problem on a mesh which is dense in the layer. The linear convergence uniform in the perturbation parameter is proved in the discrete L1 norm. Numerical results show uniform pointwise convergence too.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 1 , February 1990 , pp. 75 - 86
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Abrahamsson, L. and Osher, S., ‘Monotone difference schemes for singular perturbation problems’, SIAM J. Numer. Anal. 19 (1982), 979–992.Google Scholar
[2]Chang, K.W. and Howes, F.A., Nonlinear Singular Perturbation Phenomena: Theory and Application (Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1984).Google Scholar
[3]Liseikin, V.D. and Yanenko, N.N., ‘On the numerical solution of equations with interior and exterior boundary layers on a nonuniform mesh’, in Proc. BAIL III Conference, Editor Miller, J.J.H. (Bool Press, Dublin, 1984). pp. 68–80.Google Scholar
[4]Lorenz, J., ‘Nonlinear boundary value problems with turning points and properties of difference schemes’, in Lect. Notes in Math. 942, Editors Eckhaus, W. and de Jager, E.M. (Springer Verlag, Berlin, Heidelberg, New York, 1982). pp. 150–169.Google Scholar
[5]Lorenz, J., ‘Numerical solution of a singular perturbation problem with turning points’, in Lect. Notes in Math. 1017, Editors Knobloch, H.W. and Schmitt, K. (Springer Verlag, Berlin, New York, 1983). pp. 432–439.Google Scholar
[6]Lorenz, J., ‘Analysis of difference schemes for a stationary shock problem’, SIAM J. Numer. Anal. 21 (1984), 1038–1053.Google Scholar
[7]Niijima, K., ‘On a difference scheme of exponential type for a nonlinear singular perturbation problem’, Numer. Math. 46 (1985), 521–539.CrossRefGoogle Scholar
[8]Vulanović, R., ‘On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh’, Univ. u Novom Sadu Zb. Rad. Prirod. -Mat. Fak. Ser. Mat. 13 (1983), 187–201.Google Scholar
[9]Vulanović, R., Herceg, D. and Petrović, N., ‘On the extrapolation for a singularly perturbed boundary value problem’, Computing 36 (1986), 69–79.Google Scholar
[10]Vulanović, R., ‘A uniform numerical method for quasilinear singular perturbation problems without turning points’, Computing (to appear).Google Scholar