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Sixth-Order Adaptive Non-uniform Grids for Singularly Perturbed Boundary Value Problems

  • Sehar Iqbal
  • Paul Andries ZegelingEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)

Abstract

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.

Keywords

Boundary value problems Boundary layers Singular perturbations Adaptive non-uniform grids Optimal grids Equidistribution principle Supra-convergence 

Notes

Acknowledgement

Sehar Iqbal acknowledges the financial support by the Schlumberger Foundation (Faculty for the Future award).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUtrecht UniversityUtrechtNetherlands

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