Sixth-Order Adaptive Non-uniform Grids for Singularly Perturbed Boundary Value Problems
In this paper, a sixth order adaptive non-uniform grid has been developed for solving a singularly perturbed boundary-value problem (SPBVP) with boundary layers. For this SPBVP with a small parameter in the leading derivative, an adaptive finite difference method based on the equidistribution principle, is adopted to establish 6th order of convergence. To achieve this supra-convergence, we study the truncation error of the discretized system and obtain an optimal adaptive non-uniform grid. Considering a second order three-point central finite-difference scheme, we develop sixth order approximations by a suitable choice of the underlying optimal adaptive grid. Further, we apply this optimal adaptive grid to nonlinear SPBVPs, by using an extra approximations of the nonlinear term and we obtain almost 6th order of convergence. Unlike other adaptive non-uniform grids, our strategy uses no pre-knowledge of the location and width of the layers. We also show that other choices of the grid distributions lead to a substantial degradation of the accuracy. Numerical results illustrate the effectiveness of the proposed higher order adaptive numerical strategy for both linear and nonlinear SPBVPs.
KeywordsBoundary value problems Boundary layers Singular perturbations Adaptive non-uniform grids Optimal grids Equidistribution principle Supra-convergence
Sehar Iqbal acknowledges the financial support by the Schlumberger Foundation (Faculty for the Future award).
- 2.Ascher, U.M., Mattheij, R.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Series in Computational Mathematics, vol. 13. Prentice Hall, Englewood Cliffs (1988)Google Scholar
- 9.Degtyarev, L.M., Prozdov, V.V., Ivanova, T.S.: Mesh method for singularly perturbed one dimensional boundary value problems, with a mesh adapting to the solution. Acad. Sci. USSR 23(7), 1160–1169 (1987)Google Scholar
- 10.Eckhaus, W.: Asymptotic Analysis of Singular Perturbations. Studies in Mathematics and Its Applications, vol. 9. North-Holland, Amsterdam (1979)Google Scholar
- 23.Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994).Google Scholar
- 24.Roberts, G.O.: Computational meshes for boundary layer problems. In: Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics. Springer, Berlin (1971)Google Scholar
- 25.Schetz, J.A.: Foundations of Boundary Layer Theory for Momentum, Heat, and Mass Transfer. Prentice-Hall, Englewood Cliffs (1984)Google Scholar
- 26.Schlichting, H., Gersten, K., Krause, E., Oertel, H.: Boundary Layer Theory, vol. 7. McGraw-Hill, New York (1960)Google Scholar
- 28.Tang, T., Xu, J. (eds.): Theory and Application of Adaptive Moving Grid methods, Chapter 7. Mathematics Monograph Series: Adaptive Computations: Theory and Algorithms, vol. 6. Science Press, Beijing, (2007)Google Scholar
- 30.Verhulst, F.: Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. Texts in Applied Mathematics, vol. 50. Springer, Berlin (2005)Google Scholar