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.
This research was partially supported by the Russian Foundation for Basic Research under grant No. 16-01-00727.
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Notes
- 1.
By M (m) we denote sufficiently large (small) positive constants independent of \(\varepsilon \). In the case of grid problems, these constants also do not depend on the stencils of the difference schemes.
- 2.
The notation \(D_{(i.j)}\) (\(L_{(i.j)}\), \(m_{(i.j)}\), \(M_{(i.j)}\), \(D_{h(i.j)})\) means that these sets (operators, constants, grids) were introduced in formula (i.j).
References
Marchuk, G.: Methods of Computational Mathematics. Nauka, Moskva (1989). (in Russian)
Samarskii, A.: The Theory of Difference schemes. Nauka, Moskva (1989). (in Russian)
Kalitkin, N., Koriakin, P.: Numerical Methods. Methods of Mathematical Physics. Publishing Center “Academy”, Moskva (2013). (in Russian)
Shishkin, G.: Difference scheme for an initial-boundary value problem for a singularly perturbed transport equation. Comput. Math. Math. Phys. 57(11), 1789–1795 (2017). https://link.springer.com/article/10.1134%2FS0965542517110136
Shishkin, G.: Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural Branch, Ekaterinburg (1992). (in Russian)
Shishkin, G., Shishkina, L.: Difference Methods for Singular Perturbation Problems. Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 140. CRC Press, Boca Raton (2009)
Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer, Berlin (1996). https://doi.org/10.1007/978-3-662-03206-0
Farrel, P., Hegarty, A., Miller, J., O’Riordan, E., Shishkin, G.: Robust Computational Techniques for Boundary Layers. Chapman and Hall/CRC, Boca Raton (2000)
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Shishkina, L., Shishkin, G. (2019). Development and Numerical Study of Robust Difference Schemes for a Singularly Perturbed Transport Equation. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_55
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