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

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Numerical Geometry, Grid Generation and Scientific Computing

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 131))

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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.

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Acknowledgement

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

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Utrecht University, Utrecht, Netherlands

    Sehar Iqbal & Paul Andries Zegeling

Authors
  1. Sehar Iqbal
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  2. Paul Andries Zegeling
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Corresponding author

Correspondence to Paul Andries Zegeling .

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Editors and Affiliations

  1. Federal Research Center of Informatics and Control, Russian Academy of Sciences, Dorodnitsyn Computing Centre, Moscow, Russia

    Vladimir A. Garanzha

  2. m4sim GmbH, Berlin, Germany

    Lennard Kamenski

  3. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany

    Hang Si

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Iqbal, S., Zegeling, P.A. (2019). Sixth-Order Adaptive Non-uniform Grids for Singularly Perturbed Boundary Value Problems. In: Garanzha, V., Kamenski, L., Si, H. (eds) Numerical Geometry, Grid Generation and Scientific Computing. Lecture Notes in Computational Science and Engineering, vol 131. Springer, Cham. https://doi.org/10.1007/978-3-030-23436-2_8

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