Sixth-order adaptive non-uniform grids for singularly perturbed boundary value problems

Sehar Iqbal, Paul Andries Zegeling*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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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.

Original languageEnglish
Title of host publicationNumerical geometry, grid generation and scientific computing
Subtitle of host publicationproceedings of the 9th International Conference, NUMGRID 2018 / Voronoi 150, celebrating the 150th anniversary of G.F. Voronoi, Moscow, Russia, December 2018
EditorsVladimir A. Garanzha, Lennard Kamenski, Hang Si
PublisherSpringer
Pages109-124
Number of pages16
ISBN (Electronic)9783030234362
ISBN (Print)9783030234355
DOIs
Publication statusPublished - Jan 2019
Event9th International Conference on Numerical Geometry, Grid Generation, and Scientific Computing, celebrating the 150th anniversary of Georgy F. Voronoi, NUMGRID 2018 - Moscow, Russian Federation
Duration: 3 Dec 20185 Dec 2018

Publication series

NameLecture Notes in Computational Science and Engineering
Volume131
ISSN (Print)1439-7358
ISSN (Electronic)2197-7100

Conference

Conference9th International Conference on Numerical Geometry, Grid Generation, and Scientific Computing, celebrating the 150th anniversary of Georgy F. Voronoi, NUMGRID 2018
Country/TerritoryRussian Federation
CityMoscow
Period3/12/185/12/18

Keywords

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

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