Chebyshev-legendre super spectral viscosity method for nonlinear conservation laws

SIAM Journal on Numerical Analysis

Volumn 35, Issue 3, 1998, Pages 893-908

  Heping, M A ^a,b  

^a BROWN UNIVERSITY   (United States)

^b SHANGHAI UNIVERSITY   (China)

Author keywords

Chebyshev legendre method; Conservation laws; Convergence; Super spectral viscosity

Indexed keywords

EID: 0348024370     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0036142995293912     Document Type: Article

References (22)

1. [AGT] S. ABARBANEL, D. GOTTLIEB, AND E. TADMOR, Spectral methods for discontinuous problems, in Numerical Methods for Fluid Dynamics II, W. Morton and M. J. Baines, eds., Oxford University Press, London, 1986, pp. 129-153.
2. [CHQZ] C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.
3. [CDT] G.-Q. CHEN, Q. DU, AND E. TADMOR, Spectral viscosity approximations to multidimensional scalar conservation laws, Math. Comp., 61 (1993), pp. 629-643.
4. [D] W. S. DON, Numerical study of pseudospectral methods in shock wave applications, J. Comput. Phys., 110 (1994), pp. 103-111.
5. [DG] W. S. DON AND D. GOTTLIEB, The Chebyshev-Legendre method: Implementing Legendre methods on Chebyshev points, SIAM J. Numer. Anal., 31 (1994), pp. 1519-1534.
6. [PG1] D. FUNARO AND D. GOTTLIEB, A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations, Math. Comp., 51 (1988), pp. 599-613.
7. [FG2] D. FUNARO AND D. GOTTLIEB, Convergence results for pseudospectral approximations of hyperbolic systems by a penalty-type boundary treatment, Math. Comp., 57 (1991), pp. 585-596.
8. [GS1] D. GOTTLIEB AND C.-W. SHU, On the Gibbs phenomenon V: Recovering exponential accuracy from collocation point values of a piecewise analytic function, Numer. Math., 71 (1995), pp. 511-526.
9. [GS2] D. GOTTLIEB AND C.-W. SHU, On the Gibbs phenomenon IV: Recovering exponential accuracy in a sub-interval from the Gegenbauer partial sum of a piecewise analytic function, Math. Comp., 64 (1995), pp. 1081-1095.
10. [GS3] D. GOTTLIEB AND C.-W. SHU, On the Gibbs phenomenon 111: Recovering exponential accuracy in a sub-interval from the spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal., 33 (1996), pp. 280-290.
11. [GS4] D. GOTTLIEB AND C.-W. SHU, Private communication.
12. [GSSV] D. GOTTLIEB, C.-W. SHU, A. SOLOMONOFF, AND H. VANDEVEN, On the Gibbs phenomenon 1: Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function, J. Comput. Appl. Math., 43 (1992), pp. 81-92.
13. [GT] D. GOTTLIEB AND E. TADMOR, Recovering pointwise values of discontinuous data within spectral accuracy, in Progress and Supercomputing in Computational Fluid Dynamics, E. M. Murman and S. S. Abarbanel, eds., Birkhäuser, Boston, 1985, pp. 357-375.
14. [KZ] M. K. KWONG AND A. ZETTL, Norm inequalities for derivatives and differences, in Inequalities, Lecture Notes in Pure and Appl. Math. 129, W. N. Everitt, ed., Marcel Dekker, New York, 1991, pp. 91-121.
15. [M] H.-P. MA, Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 869-892.
16. [MOT] Y. MADAY, S. M. OULD KABER, AND E. TADMOR, Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 30 (1993), pp. 321-342.
17. [MT] Y. MADAY AND E. TADMOR, Analysis of the spectral viscosity method for periodic conservation laws, SIAM J. Numer. Anal., 26 (1989), pp. 854-870.
18. [SW] C.-W. SHU AND P. WONG, A note on the accuracy of spectral method applied to nonlinear conservation laws, J. Sci. Comput., 10 (1995), pp. 357-369.
19. [Ta1] E. TADMOR, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26 (1989), pp. 30-44.
20. [Ta2] E. TADMOR, Shock capturing by the spectral viscosity method, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 197-208.
21. [Ta3] E. TADMOR, Super viscosity and spectral approximations of nonlinear conservation laws, in Numerical Methods for Fluid Dynamics IV, M. J. Baines and K. W. Morton, eds., Clarendon Press, Oxford, 1993, pp. 69-82.
22. [Tr] L. TARTAR, Compensated compactness and applications to partial differential equations, in Res. Notes in Math. 39, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. 4, R. J. Knopp, ed., Pitman Press, Boston, London, 1975, pp. 136-211.