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The Dissipative Spectral Methods for the First Order Linear Hyperbolic Equations

Published online by Cambridge University Press:  28 May 2015

Lian Chen ,
Zhongqiang Zhang  and
Heping Ma
Lian Chen*
Affiliation:
Department of Mathematics, College of Sciences, Shanghai University, Shanghai, 200444, China
Zhongqiang Zhang*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, Rhode Island, 02912, USA
Heping Ma*
Affiliation:
Department of Mathematics, College of Sciences, Shanghai University, Shanghai, 200444, China
*
Corresponding author.Email address:chenlianice@163.com
Corresponding author.Email address:handyzang@gmail.com
Corresponding author.Email address:hpma@shu.edu.cn
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Abstract

In this paper, we introduce the dissipative spectral methods (DSM) for the first order linear hyperbolic equations in one dimension. Specifically, we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition. The error estimates of the methods are shown to be quasi-optimal for variable-coefficients equations. Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods, showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.

Keywords

65N1565N35First order hyperbolic equationdissipative spectral methoderror estimate
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 3 , August 2012 , pp. 493 - 508
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Aliabadi, M. H. and Ortiz, E. L., Numerical treatment of moving and free boundary value problems with the tau method, Computers Math. Applic., vol. 35, no. 8 (1998), pp. 53–61.CrossRefGoogle Scholar
[2]Canuto, C., Hussaini, M. Y., Quartesoni, A., and Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer, 2006.CrossRefGoogle Scholar
[3]Charalambides, M. and Waleffe, F., Spectrum of the Jacobi tau approximation for the second derivative operator, SIAM J. Numer. Anal., 46 (2008), pp. 280–294.CrossRefGoogle Scholar
[4]Cui, K. and Ma, H. P., The Legendre-tau method for a first-order hyperbolic equation(In Chinese), J. Commun. Appl. Math. Comput., 23 (2009), pp. 42–52.Google Scholar
[5]Cui, Y. F. and Mao, D. K., Numerical method satisfying the first two conservation laws for the Kortewegde Vries equation, J. Comput. Phys., 227 (2007), pp. 376–399.CrossRefGoogle Scholar
[6]Dawkins, P. T., Dunbar, S. R., and Douglass, R. W., The origin and nature of spurious eigenvalues in the spectral tau method, J. Comput. Phys., 147 (1998), pp. 441–462.CrossRefGoogle Scholar
[7]Dendy, J. E., Two methods of Galerkin type achieving optimum L2 rates of convergence for first order hyperbolics, SIAM J. Numer. Anal., 11 (1974), pp. 637–653.CrossRefGoogle Scholar
[8]El-Daou, M. K. and Ortiz, E. L., Error analysis of the tau method: Dependence of the error on the degree and the length of the interval of approximation, Computers Math. Applic., vol. 25, no. 7 (1993), pp. 33–45.CrossRefGoogle Scholar
[9]El-Daou, M. K. and Ortiz, E. L., A posteriori error bounds for the approximate solution of second-order ODEs by piecewise coefficients perturbation methods, J. Comput. Appl. Math., 189 (2006), pp. 51–66.CrossRefGoogle Scholar
[10]Gottlieb, D. and Hesthaven, J. S., Spectral methods for hyperbolic problems, J. Comput. Appl. Math., 128 (2001), pp. 83–131.CrossRefGoogle Scholar
[11]Huang, W. Z., Ma, H. P., and Sun, W. W., Convergence analysis of spectral collocation methods for a singular differential equation, SIAM J. Numer. Anal., 41 (2004), pp. 2333–2349.Google Scholar
[12]Hughes, T. J. and Brooks, A. N., A multi-dimensional upwind scheme with no crosswind diffusion, in Finite Element Methods for Convection Dominated Flows, Hughes, T. J., ed., vol. 34, ASME, New York, 1979, pp. 19–35.Google Scholar
[13]Kreiss, H. O. and Oliger, J., Stability of the Fourier method, SIAM J. Numer. Anal., 16 (1979), pp. 421–433.CrossRefGoogle Scholar
[14]Landriani, G. S., Spectral tau approximation of the two-dimensional Stokes problem, Numer. Math., 52 (1988), pp. 683–699.CrossRefGoogle Scholar
[15]LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhauser-verlag, 1992.CrossRefGoogle Scholar
[16]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge Univ. Press., 2002.Google Scholar
[17]Li, H. X., Wang, Z. G., and Mao, D. K., Numerically neither dissipative nor compressive scheme for linear advection equation and its application to the Euler system, J. Sci. Comput., 36 (2008), pp. 285–331.CrossRefGoogle Scholar
[18]Nitsche, J. A., Ein kriterium für die quasi-optimalitat des Ritzchen Verfahrens, Numer. Math., 11 (1968), pp. 346–348.CrossRefGoogle Scholar
[19]Shen, J., A spectral-tau approximation for the Stokes and Navier-Stokes equations, Math. Model. Num. Anal., 22 (1988), pp. 677–693.CrossRefGoogle Scholar
[20]Shen, J., A new dual-Petrov–Galerkin method for third and higher odd-order differential equations: Application to the KdV equation, SIAM J. Numer. Anal., 41 (2004), pp. 1595–1619.Google Scholar
[21]Shen, J. and Wang, L. L., Legendre and Chebyshev dual-Petrov–Galerkin methods for hyperbolic equations, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 3785–3797.CrossRefGoogle Scholar
[22]Shen, T. T., Zhang, Z. Q., and Ma, H. P., Optimal error estimates of the Legendre tau method for second-order differential equations, J. Sci. Comput., 42 (2010), pp. 198–215.CrossRefGoogle Scholar
[23]Stynes, M. and Tobiska, L., The SDFEM for a convection-diffusion problem with a boundary layer: Optimal error analysis and enhancement of accuracy, SIAM J. Numer. Anal., 41 (2004), pp. 1620–1642.Google Scholar
[24]Tang, J. G. and Ma, H. P., Single and multi-interval Legendre τ-methods in time for parabolic equations, Adv. Comput. Math., 17 (2002), pp. 349–367.CrossRefGoogle Scholar
[25]Wahlbin, L. B., A dissipative Galerkin method applied to some quasilinear hyperbolic equations, Revue francaise d’automatique, informatique, recherche opérationelle. Analyse numérique, 8 (1974), pp. 109–117.Google Scholar
[26]Wahlbin, L. B., A dissipative Galerkin method for the numerical solution of first-order hyperbolic equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, 1974, pp. 147–169.Google Scholar
[27]Wang, Z. G. and Mao, D. K., A finite difference scheme for linear advection equation satisfying three conservation laws (in Chinese), J. Shanghai Univ. Nat. Sci., vol. 12, no. 6 (2006), pp. 588–598.Google Scholar
[28]Yang, Z. and Zhang, Z. Q., An optimal error estimate of dissipative spectral tau method for a first order hyperbolic equation (in Chinese), Journal of Lishui University, vol. 30, no. 5 (2008), pp. 12–15.Google Scholar
[29]Zhao, J. M. and Liu, L. H., Spectral element method with adaptive artificial diffusion for solving the radiative transfer equation, Numerical Heat Transfer, Part B: Fundamentals, 53 (2008), pp. 536–554.CrossRefGoogle Scholar
[30]Zhou, G. H., How accurate is the streamline diffusion finite element method? Math. Comp., 66 (1997), pp. 31–44.CrossRefGoogle Scholar