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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 3/2018

11.07.2017

Enriched Spectral Method for Stiff Convection-Dominated Equations

verfasst von: Youngjoon Hong, Chang-Yeol Jung

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2018

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Abstract

A novel and simple numerical method for stiff convection-dominated problems is studied in presence of boundary or interior layers. A version of the spectral Chevyshev-collocation method enriched with the so-called corrector functions is investigated. The corrector functions here are designed to capture the stiffness of the layers (see the Appendix), and the proposed method does not rely on the adaptive grid points. The extensive numerical results demonstrate that the enriched spectral methods are very accurate with low computational cost.
Vorheriger Artikel A Homotopy Method for Parameter Estimation of Nonlinear Differential Equations with Multiple Optima
Nächster Artikel Simulating Compressible Two-Medium Flows with Sharp-Interface Adaptive Runge–Kutta Discontinuous Galerkin Methods

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Literatur
1.
Alexander, N.B., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3):199–259 (1982). FENOMECH ’81, Part I (Stuttgart, 1981)
2.
Berger, A., De Han, H., Kellogg, B.: A priori estimates and analysis of a numerical method for a turning point problem. Math. Comput. 42(166), 465–492 (1984)MathSciNetCrossRefMATH
3.
4.
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, New York (1988)CrossRefMATH
5.
Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Spence, E.A.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. 21, 89–305 (2012)MathSciNetCrossRefMATH
6.
DeSanti, A.J.: Nonmonotone interior layer theory for some singularly perturbed quasilinear boundary value problems with turning points. SIAM J. Math. Anal. 18(2), 321–331 (1987)MathSciNetCrossRefMATH
7.
DeSanti, A.J.: Perturbed quasilinear Dirichlet problems with isolated turning points. Commun. Partial Differ. Equ. 12(2), 223–242 (1987)MathSciNetCrossRefMATH
8.
de Groen, P.P.N., Hemker, P.W.: Error bounds for exponentially fitted Galerkin methods to stiff boundary value problems. In: Numerical and Asymptotical Methods for Singular Perturbation Problems, pp. 217–249. Academic Press, New York (1979)
9.
De Han, H., Huang, Z.: Tailored finite point method for steady-state reaction–diffusion equations. Commun. Math. Sci. 8(4), 887–899 (2013)MathSciNetCrossRefMATH
10.
De Han, H., Huang, Z.: Tailored finite point method based on exponential bases for convection–diffusion–reaction equation. Math. Comput. 82(281), 213–226 (2010)MathSciNetCrossRefMATH
11.
De Han, H., Kellogg, B.: The use of enriched subspaces for singular perturbation problems. In: Proceedings of the China-France Symposium on Finite Element Methods (Beijing, 1982), pp. 293–305. Sci. Press Beijing, Beijing (1983)
12.
13.
de Ruijter, W.P.M.: On the Asymptotic Analysis of Large-Scale Ocean Circulation, volume 120 of Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam (1980)MATH
14.
Eisen, H., Heinrichs, W.: A new method of stabilization for singular perturbation problems with spectral methods. SIAM J. Numer. Anal. 29(1), 107–122 (1992)MathSciNetCrossRefMATH
15.
Gie, G.-M., Jung, C.-Y., Temam, R.: Recent progresses in boundary layer theory. Discrete Contin. Dyn. Syst. 36(5), 2521–2583 (2016)MathSciNetMATH
16.
Graciaa, J.L., O’Riordan, E.: Numerical approximation of solution derivatives in the case of singularly perturbed time dependent reaction-diffusion problems. J. Comput. Appl. Math. 273(1), 13–24 (2015)MathSciNetCrossRefMATH
17.
Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (1977). CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26
18.
Guo, W., Stynes, M.: Pointwise error estimates for a streamline diffusion method on a Shishkin mesh for a convection–diffusion problem. IMA J. Numer. Anal. 17(1), 29–59 (1997)MathSciNetCrossRefMATH
19.
Hong, Y., Jung, C.-Y., Temam, R.: On the numerical approximations of stiff convection–diffusion equations in a circle. Numer. Math. 127(2), 291–313 (2014)MathSciNetCrossRefMATH
20.
Hong, Y.: Numerical approximation of the singularly perturbed heat equation in a circle. J. Sci. Comput. 62(1), 1–24 (2015)MathSciNetCrossRefMATH
21.
Il’in, A.M.: A difference scheme for a differential equation with a small parameter affecting the highest derivative (in Russian). Mat. Zametki. 6, 237–248 (1969)MathSciNet
22.
John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations. I. A review. Comput. Methods Appl. Mech. Eng. 196(17–20), 2197–2215 (2007)MathSciNetCrossRefMATH
23.
John, V., Knobloch, P., Savescu, S.B.: A posteriori optimization of parameters in stabilized methods for convection–diffusion problems—Part I. Comput. Methods Appl. Mech. Eng. 200(41–44), 2916–2929 (2011)MathSciNetCrossRefMATH
24.
Jung, C.-Y., Nguyen, T.B.: Semi-analytical numerical methods for convection-dominated problems with turning points. Int. J. Numer. Anal. Model. 10(2), 314–332 (2013)MathSciNetMATH
25.
Jung, C.-Y., Temam, R.: Asymptotic analysis for singularly perturbed convection–diffusion equations with a turning point. J. Math. Phys. 48(6), 065301 (2007)MathSciNetCrossRefMATH
26.
Jung, C.-Y., Temam, R.: Finite volume approximation of one-dimensional stiff convection–diffusion equations. J. Sci. Comput. 41(3), 384–410 (2009)MathSciNetCrossRefMATH
27.
Jung, C.-Y., Temam, R.: Finite volume approximation of two-dimensional stiff problems. Int. J. Numer. Anal. Model. 7(3), 462–476 (2010)MathSciNetMATH
28.
Jung, C.-Y., Temam, R.: Singularly perturbed problems with a turning point: the non-compatible case. Anal. Appl. (Singap.) 12(3), 293–321 (2014)MathSciNetCrossRefMATH
29.
Jung, J.-H.: A note on the spectral collocation approach of some differential equations with singular source terms. J. Sci. Comput. 39, 49–66 (2009)MathSciNetCrossRefMATH
30.
Kieri, E.: Accelerated convergence for Schrödinger equations with non-smooth potentials. BIT Numer. Math. 54, 729–748 (2014)MathSciNetCrossRefMATH
31.
Kellogg, B., Linss, T., Stynes, M.: A finite difference method on layer-adapted meshes for an elliptic reaction–diffusion system in two dimensions. Math. Comput. 77, 2085–2096 (2008)MathSciNetCrossRefMATH
32.
Kellogg, R.B., Xenophontos, C.: An enriched subspace finite element method for convection–diffusion problems. Int. J. Numer. Anal. Model. 7(3), 477–490 (2010)MathSciNetMATH
33.
Li, Z.-C., Hu, H.-Y., Hsu, C.-H., Wang, S.: Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains. I. Convergence analysis. J. Comput. Appl. Math. 166, 181–208 (2004)MathSciNetCrossRefMATH
34.
35.
Liu, W.B., Shen, J.: A new efficient spectral Galerkin method for singular perturbation problems. J. Sci. Comput. 11(4), 411–437 (1996)MathSciNetCrossRefMATH
36.
Liu, W., Tang, T.: Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems. Appl. Numer. Math. 38(3), 315–345 (2001)MathSciNetCrossRefMATH
37.
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific Publishing Co., Inc., River Edge (1996)CrossRefMATH
38.
39.
Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations, volume 24 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2008). Convection–diffusion–reaction and flow problems
40.
Simonnet, E., Dijkstra, H.A.: Spontaneous generation of low-frequency modes of variability in the wind-driven ocean circulation. J. Phys. Oceanogr. 32, 1747–1762 (2002)CrossRef
41.
Smith, D.R.: Singular-Perturbation Theory. Cambridge University Press, Cambridge (1985). An introduction with applications
42.
Shen, J., Tang, T., Wang, L.-L.: Spectral Methods, Volume 41 of Springer Series in Computational Mathematics. Springer, Heidelberg (2011). Algorithms, analysis and applications
43.
Shen, J., Wang, Y.: Müntz–Galerkin methods and applications to mixed Dirichlet–Neumann boundary value problems. SIAM J. Sci. Comput. 38(4), A2357 (2016)CrossRefMATH
44.
Shih, S.-D., Kellogg, R.B.: Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal. 18(5), 1467–1511 (1987)MathSciNetCrossRefMATH
45.
Trefethen, L.N.: Spectral Methods in MATLAB, Volume 10 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)CrossRef
46.
Tang, T., Trummer, M.R.: Boundary layer resolving pseudospectral methods for singular perturbation problems. SIAM J. Sci. Comput. 17(2), 430–438 (1996)MathSciNetCrossRefMATH
47.
Tee, T.W., Trefethen, L.N.: A rational spectral collocation method with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28(5), 1798–1811 (2006). (electronic)MathSciNetCrossRefMATH
48.
Wang, Y., Chen, S., Wu, X.: A rational spectral collocation method for solving a class of parameterized singular perturbation problems. J. Comput. Appl. Math. 233(10), 2652–2660 (2010)MathSciNetCrossRefMATH
49.
Zrahia, U., Orszag, S.A., Israeli, M.: Hybrid spectral element/asymptotic method for boundary layers problems. J. Comput. Phys. 138(2), 858–880 (1997)MathSciNetCrossRefMATH
Metadaten
Titel
Enriched Spectral Method for Stiff Convection-Dominated Equations
verfasst von
Youngjoon Hong
Chang-Yeol Jung
Publikationsdatum
11.07.2017
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-017-0494-8

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