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Development and Numerical Study of Robust Difference Schemes for a Singularly Perturbed Transport Equation

  • Lidia ShishkinaEmail author
  • Grigorii Shishkin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

On the set \(\overline{G} =G \cup S\), \(G=(0,d]\times (0,T]\) with the boundary \(S=S_0 \cup S^{\,\ell }\), we consider an initial-boundary value problem for the singularly perturbed transport equation with a perturbation parameter \(\varepsilon \) multiplying the spatial derivative, \(\varepsilon \in (0,1]\). For small values of the perturbation parameter \(\varepsilon \), the solution of such a problem has a singularity of the boundary layer type, which makes standard difference schemes unsuitable for practical computations. To solve this problem numerically, an approach to the development of a robust difference scheme is proposed, similar to that used for constructing special \(\varepsilon \)-uniformly convergent difference schemes for singularly perturbed elliptic and parabolic equations. In this paper, we give a technique for constructing a robust difference scheme and justifying its \(\varepsilon \)-uniform convergence, and we study numerically solutions of standard and special robust difference schemes for a model initial-boundary value problem for a singularly perturbed transport equation. The results of numerical experiments confirm theoretical results.

Keywords

Singularly perturbed transport equation Boundary layer Robust difference scheme \(\varepsilon \)-uniform convergence Maximum norm Standard difference scheme Uniform mesh Piecewise-uniform mesh Solution decomposition A priori estimates 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics, UB RASYekaterinburgRussia

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