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Total variation and error estimates for spectral viscosity approximations


Author: Eitan Tadmor
Journal: Math. Comp. 60 (1993), 245-256
MSC: Primary 35L65; Secondary 65M06, 65M12, 65M15
DOI: https://doi.org/10.1090/S0025-5718-1993-1153170-9
MathSciNet review: 1153170
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Abstract: We study the behavior of spectral viscosity approximations to non-linear scalar conservation laws. We show how the spectral viscosity method compromises between the total-variation bounded viscosity approximations-- which are restricted to first-order accuracy--and the spectrally accurate, yet unstable, Fourier method. In particular, we prove that the spectral viscosity method is $ {L^1}$-stable and hence total-variation bounded. Moreover, the spectral viscosity solutions are shown to be $ {\text{Lip}^ + }$-stable, in agreement with Oleinik's E-entropy condition. This essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution, and we provide convergence rate estimates of both global and local types.


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  • [1] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. Zang, Spectral methods in fluid dynamics, Springer-Verlag, New York, 1988. MR 917480 (89m:76004)
  • [2] D. Gottlieb and S. Orszag, Numerical analysis of spectral methods: Theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics 25, SIAM, Philadelphia, PA, 1977. MR 0520152 (58:24983)
  • [3] D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in "Progress in Supercomputing in Computational Fluid Dynamics," Progress in Scientific Computing, Vol. 6 (E. M. Murman and S. S. Abarbanel, eds.), Birkhäuser, Boston, 1985, pp. 357-375. MR 935160 (90a:65041)
  • [4] H.-O. Kreiss, Fourier expansions of the solutions of Navier-Stokes equations and their exponential decay rate, Analyse Mathématique et Appl., Gauthier-Villars, Paris, 1988, pp. 245-262. MR 956963 (89k:35184)
  • [5] H.-O. Kreiss and J. Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 (1972), 199-215. MR 0319382 (47:7926)
  • [6] N. N. Kuznetsov, On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions, Topics in Numerical Analysis III (Proc. Roy. Irish Acad. Conf., J. J. H. Miller, ed.), Academic Press, London, 1977, pp. 183-197. MR 0657786 (58:31874)
  • [7] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conf. Series in Appl. Math., SIAM, Philadelphia, PA, 1973. MR 0350216 (50:2709)
  • [8] Y. Maday and E. Tadmor, Analysis of the spectral viscosity method for periodic conservation laws, SIAM J. Numer. Anal. 26 (1989), 854-870. MR 1005513 (90f:65153)
  • [9] H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear conservation laws, SIAM J. Numer. Anal. 29 (1992), in press. MR 1191133 (93j:65139)
  • [10] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91-106. MR 679435 (84a:65075)
  • [11] S. Schochet, The rate of convergence of spectral-viscosity methods for periodic scalar conservation laws, SIAM J. Numer. Anal. 27 (1990), 1142-1159. MR 1061123 (91g:65207)
  • [12] J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1983. MR 688146 (84d:35002)
  • [13] E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal. 26 (1989), 30-44. MR 977947 (90e:65130)
  • [14] -, Shock capturing by the spectral viscosity method, Comput. Methods Appl. Mech. Engrg. 78 (1990), 197-208. MR 1067951 (91g:35174)
  • [15] -, Semi-discrete approximations to nonlinear systems conservation laws; consistency and $ {L^\infty }$-stability imply convergence, ICASE Report No. 88-41.
  • [16] -, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891-906. MR 1111445 (92d:35190)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1153170-9
Keywords: Conservation laws, spectral viscosity method, spectral accuracy, total variation, Lipschitz stability, convergence rate estimates
Article copyright: © Copyright 1993 American Mathematical Society

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