Back
 JAMP  Vol.8 No.10 , October 2020
Numerical Solution of Quasilinear Singularly Perturbed Problems by the Principle of Equidistribution
Abstract:
In this paper, the numerical solution and its error analysis of quasilinear singular perturbation two-point boundary value problems based on the principle of equidistribution are given. On the non-uniform grid of the uniformly distributed arc-length monitor function, the solution of the simple upwind scheme is obtained. It is proved that the adaptive simple upwind scheme based on the principle of equidistribution has uniform convergence for small perturbation parameters. Numerical experiments are carried out and the error analysis are confirmed.
Cite this paper: Zheng, Q. and Ye, F. (2020) Numerical Solution of Quasilinear Singularly Perturbed Problems by the Principle of Equidistribution. Journal of Applied Mathematics and Physics, 8, 2175-2181. doi: 10.4236/jamp.2020.810163.
References

[1]   Tollmien, W., Schlichting, H., Grtler, H. and Riegels, F.W. (1961) über Flüssigkeitsbewegung bei sehr kleiner Reibung. Ludwig Prandtl Gesammelte Abhandlungen. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11836-8

[2]   Roos, H.-G., Stynes, M. and Tobiska, L. (2008) Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, Heidelberg.

[3]   Kopteva, N.V. (1997) On the Uniform in Small Parameter Convergence of a Weighted Scheme for the One-Dimensional Time-Dependent Convection-Diffusion Equation. Computational Mathematics and Mathematical Physics, 37, 1173-1180.

[4]   Linβ, T. (2001) Uniform Pointwise Convergence of Finite Difference Schemes Using Grid Equidistribution. Computing, 66, 27-39. https://doi.org/10.1007/s006070170037

[5]   Kopteva, N. and Stynes, M. (2001) A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem. SIAM Journal on Numerical Analysis, 39, 1446-1447. https://doi.org/10.1137/S003614290138471X

[6]   Jugal, M. and Srinivasan, N. (2010) Parameter-Uniform Numerical Method for Global Solution and Global Normalized Flux of Singularly Perturbed Boundary Value Problems Using Grid Equi-distribution. Computers & Mathematics with Applications, 60, 1924-1939. https://doi.org/10.1016/j.camwa.2010.07.026

[7]   Linβ, T., Roos, H.G. and Vulanovic, R. (2000) Uniform Pointwise Convergence on Shishkin-Type Meshes for Quasi-Linear Convection-Diffusion Problems. SIAM Journal on Numerical Analysis, 38, 897-912. https://doi.org/10.1137/S0036142999355957

[8]   Linβ, T. (2001) Sufficient Conditions for uniform Convergence on Layer-Adapted Grids. Applied Numerical Mathematics, 37, 241-255. https://doi.org/10.1016/S0168-9274(00)00043-X

[9]   Linβ, T. (2001) Uniform Second-Order Pointwise Convergence of a Finite Difference Discretization for a Quasilinear Problem. Computational Mathematics & Mathematical Physics, 41, 947-958.

[10]   Zheng, Q., Li, X. and Gao, Y. (2015) Uniformly Convergent Hybrid Schemes for Solutions and Derivatives in Quasilinear Singularly Perturbed BVPs. Applied Numerical Mathematics, 91, 46-59. https://doi.org/10.1016/j.apnum.2014.12.010

[11]   Vulanovic, R. and Hovhannisyan, G. (2006) A Posteriori Error Estimates for One- Dimensional Convection-Diffussion Problems. Computers & Mathematics with Applications, 51, 915-926. https://doi.org/10.1016/j.camwa.2005.11.027

 
 
Top