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Filtering in Legendre spectral methods
Author(s):
Jan
S.
Hesthaven;
Robert
M.
Kirby.
Journal:
Math. Comp.
77
(2008),
1425-1452.
MSC (2000):
Primary 65M70;
Secondary 65M12
Posted:
March 5, 2008
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Abstract:
We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments.
References:
-
- 1.
- C. BERNARDI AND Y. MADAY, Polynomial Interpolation Results in Sobolev Spaces, J. Comput. Appl. Math. 43(1992), pp. 53-80. MR 1193294 (93k:65010)
- 2.
- J.P. BOYD, Two Comments on Filtering (Artificial Viscosity) for Chebyshev and Legendre Spectral and Spectral Element Methods: Preserving Boundary Conditions and Interpretation of the Filter as a Diffusion, J. Comput. Phys. 142(1998), pp. 283-288. MR 1624716
- 3.
- C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer-Verlag. New York, 1988. MR 917480 (89m:76004)
- 4.
- C. CANUTO AND A. QUARTERONI, Approximation Results for Orthogonal Polynomials in Sobolev Spaces, Math. Comp. 38(1982), pp. 67-86. MR 637287 (82m:41003)
- 5.
- W. S. DON, Numerical Study of Pseudospectral Methods in Shock Wave Applications, J. Comput. Phys. 110(1994), pp. 103-111.
- 6.
- W. S. DON AND D. GOTTLIEB, Spectral Simulation of Supersonic Reactive Flows, SIAM J. Numer. Anal. 35(1998), pp. 2370-2384. MR 1655851 (99i:65110)
- 7.
- A. ERD´ELYI (EDS), Higher Transcendental Functions, Vol II. Robert E. Krieger Publishing Company, Florida, 1981. MR 698780 (84h:33001b)
- 8.
- M.O. DEVILLE, P.F. FISCHER, AND E.H. MUND, High-Order Methods for Incompressible Fluid Flow, Cambridge University Press, 2002. MR 1929237 (2003g:76071)
- 9.
- P.F. FISCHER AND J.S. MULLEN, Filter-Based Stabilization of Spectral Element Methods, C. R. Acad. Sci. Paris 332(2001), pp. 265-270 (2001). MR 1817374 (2001m:65129)
- 10.
- D. GOTTLIEB, S.A. ORSZAG, AND E. TURKEL, Stability of Pseudospectral and Finite-Difference Methods for Variable Coefficient Problems, Math. Comp. 37(1981), pp. 293-305. MR 628696 (82i:65054)
- 11.
- D. GOTTLIEB AND E. TADMOR, Recovering Pointwise Values of Discontinuous Data with Spectral Accuracy. In Progress and Supercomputing in Computational Fluid Dynamics. Birkhäuser, Boston, 1984. pp. 357-375. MR 935160 (90a:65041)
- 12.
- D. GOTTLIEB AND C. W. SHU, On the Gibbs Phenomenon and its Resolution, SIAM Review 39(1997), pp. 644-668. MR 1491051 (98m:42002)
- 13.
- D. GOTTLIEB AND J. S. HESTHAVEN, Spectral Methods for Hyperbolic Problems, J. Comp. Appl. Math. 128(2001), pp. 83-131. MR 1820872 (2001m:65138)
- 14.
- J. S. HESTHAVEN AND D. GOTTLIEB, A Stable Penalty Method for the Compressible Navier-Stokes Equations. I. Open Boundary Conditions, SIAM J. Sci. Comp. 17(1996), 579-612. MR 1384253 (97j:65142)
- 15.
- J. S. HESTHAVEN, Spectral Penalty Methods, Appl. Numer. Math. 23(2000), pp. 23-41. MR 1770238 (2001f:65118)
- 16.
- J. S. HESTHAVEN, S. GOTTLIEB, AND D. GOTTLIEB, Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge, UK, 2007. MR 2333926
- 17.
- A KANEVEKY, M.H. CARPENTER, AND J.S. HESTHAVEN, Idempotent Filtering in Spectral and Spectral Element Methods, J. Comput. Phys. 220(2006), pp. 41-58. MR 2281620 (2007k:65155)
- 18.
- G.E. KARNIADAKIS AND S. J. SHERWIN, Spectral/hp Element Methods for CFD, Oxford University Press, Oxford, UK, 1999. MR 1696933 (2000h:76120)
- 19.
- R.M. KIRBY AND G. E. KARNIADAKIS, De-aliasing on Non-uniform Grids: Algorithms and Applications, J. Comput. Phys. 191(2003) pp. 249-264
- 20.
- D.A. KOPRIVA, A Practical Assessment of Spectral Accuracy for Hyperbolic Problems with Discontinuities, J. Sci. Comput. 2(1987), pp. 249-262.
- 21.
- H.O. KREISS AND J. OLIGER, Stability of the Fourier Method, SIAM J. Numer. Anal. 16(1979), pp. 421-433. MR 530479 (80i:65130)
- 22.
- Y. MADAY AND E. TADMOR, Analysis of the Spectral Vanishing Viscosity Method for Periodic Conservation Laws, SIAM J. Numer. Anal. 26(1989), pp. 854-870. MR 1005513 (90f:65153)
- 23.
- Y. MADAY, S. M. OULD KAPER, AND E. TADMOR, Legendre Pseudospectral Viscosity Method for Nonlinear Conservation Laws, SIAM J. Numer. Anal. 30(1993), pp. 321-342. MR 1211394 (93m:65148)
- 24.
- R. PASQUETTI AND C. J. XU, Comments on ``Filter-Based Stabilization of Spectral Element Methods'', J. Comput. Phys. 182(2002), pp. 646-650. MR 1941853 (2003k:76096)
- 25.
- E. TADMOR, Convergence of Spectral Methods for Nonlinear Conservation Laws, SIAM J. Numer. Anal. 26(1989), pp. 30-44. MR 977947 (90e:65130)
- 26.
- E. TADMOR AND J. TANNER, Adaptive mollifiers - High Resolution Recovery of Piecewise Smooth Data from its Spectral Information, Foundat. Comput. Math. 2(2002), pp. 155-189. MR 1894374 (2003b:42009)
- 27.
- H. VANDEVEN, Family of Spectral Filters for Discontinuous Problems, J. Scient. Comput. 6(1991), pp. 159-192. MR 1140344 (92k:65006)
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Additional Information:
Jan
S.
Hesthaven
Affiliation:
Division of Applied Mathematics, Brown University, Box F, Providence, Rhode Island 02912
Email:
Jan.Hesthaven@brown.edu
Robert
M.
Kirby
Affiliation:
School of Computing, University of Utah, Salt Lake City, Utah 84112
Email:
kirby@cs.utah.edu
DOI:
10.1090/S0025-5718-08-02110-8
PII:
S 0025-5718(08)02110-8
Keywords:
Spectral methods,
filtering,
stabilization,
Legendre polynomials
Received by editor(s):
January 2, 2004
Received by editor(s) in revised form:
July 21, 2004
Posted:
March 5, 2008
Additional Notes:
The work of the first author was partly supported by NSF Career Award DMS-0132967, NSF International Award INT-0307475, ARO under contract DAAD19-01-1-0631, and the Alfred P. Sloan Foundation through a Sloan Research Fellowship.
The work of the second author was supported by NSF Career Award NSF-CCF0347791.
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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