A pseudospectral fictitious point method for high order initial-boundary value problems

SIAM Journal on Scientific Computing

Volumn 28, Issue 5, 2006, Pages 1716-1729

  Fornberg, Bengt ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

Kuramoto Sivashinsky equation; PDE; Pseudospectral method; Spurious eigenvalues; Time space corner singularities

Indexed keywords

APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; PARTIAL DIFFERENTIAL EQUATIONS;

KURAMOTO-SIVASHINSKY EQUATION; PSEUDOSPECTRAL METHOD; SPURIOUS EIGENVALUES; TIME-SPACE CORNER SINGULARITIES;

SPECTRUM ANALYSIS;

EID: 34648828036     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040611252     Document Type: Article

References (38)

1. U. M. ASCHER, S. J. RUUTH, AND S. T. R. WETTON, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), pp. 797-823.
2. G. BEYLKIN, J. M. KEISER, AND L. VOZOVOI, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys., 147 (1998), pp. 362-387.
3. J. P. BOYD, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover, New York, 2000.
4. J. P. BOYD AND N. FLYER, Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 281-309.
5. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988.
6. S. M. COX AND P. C. MATTHEWS, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430-455.
7. T. A. DRISCOLL AND B. FORNBERG, A block pseudospectral method for Maxwell's equations: I. One-dimensional case, J. Comput. Phys., 140 (1998), pp. 47-65.
8. T. A. DRISCOLL AND B. FORNBERG, Block pseudospectral methods for Maxwell's equations II: Two-dimensional, discontinuous-coefficient case, SIAM J. Sci. Comput., 21 (1999), pp. 1146-1167.
9. N. FLYER AND B. FORNBERG, Accurate numerical resolution of transients in initial-boundary value problems for the heat equation, J. Comput. Phys., 184 (2003), pp. 526-539.
10. N. FLYER AND B. FORNBERG, On the nature of initial-boundary value solutions for dispersive equations, SIAM J. Appl. Math., 64 (2003), pp. 546-564.
11. N. FLYER AND P. N. SWARZTRAUBER, The convergence of spectral and finite difference methods for initial-boundary value problems, SIAM J. Sci. Comput., 23 (2002), pp. 1731-1751.
12. B. FORNBERG, Generation of finite difference formulas on arbitrarily spaced grids, Math. Comp., 51 (1988), pp. 699-706.
13. B. FORNBERG, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, UK, 1996.
14. B. FORNBERG, Calculation of weights in finite difference formulas, SIAM Rev., 40 (1998), pp. 685-691.
15. D. R. GARDNER, S. A. TROGDON, AND R. W. DOUGLAS, A modified tau spectral method that eliminates spurious eigenvalues, J. Comput. Phys., 80 (1989), pp. 137-167.
16. D. FUNARO AND W. HEINRICHS, Some results about the pseudospectral approximation of one-dimensional fourth-order problems, Numer. Math., 58 (1990), pp. 399-418.
17. W. HEINRICHS, Spectral approximation of third-order problems, J. Sci. Comput., 14 (1999), pp. 275-289.
18. W. HUANG AND D. M. SLOAN, The pseudospectral method for third-order differential equations, SIAM J. Numer. Anal., 29 (1992), pp. 1626-1647.
19. W. HUANG AND D. M. SLOAN, The pseudospectral method for solving differential eigenvalue problems, J. Comput. Phys., 111 (1994), pp. 399-409.
20. J. M. HYMAN AND B. NICOLAENKO, The Kuramoto-Sivashinsky equation: A bridge between partial differential equations and dynamical systems, Phys. D, 18 (1986), pp. 113-126.
21. D. A. JONES, L. G. MARGOLIN, AND A. C. POJE, Enslaved finite difference schemes for non-linear dissipative PDEs, Numer. Methods Partial Differential Equations, 12 (1996), pp. 13-40.
22. A.-K. KASSAM AND L. N. TREFETHEN, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214-1233.
23. G. B. MCFADDEN, B. T. MURRAY, AND R. F. BOISVERT, Elimination of spurious eigenvalues in the Chebyshev tau spectral method, J. Comput. Phys., 91 (1990), pp. 228-239.
24. W. J. MERRYFIELD AND B. SHIZGAL, Note: Properties of collocation third-derivative operators, J. Comput. Phys., 105 (1993), pp. 182-185.
25. D. R. MOTT, E. S. ORAN, AND B. VAN LEER, A quasi-steady state solver for the stiff ordinary differential equations of reaction kinetics, J. Comput. Phys., 164 (2000), pp. 407-428.
26. L. S. MULHOLLAND, The Eigenvalues of Third-Order Chebyshev Pseudospectral Differentiation Matrices, Mathematics Research Report, 6, University of Strathclyde, Glasgow, UK, 1995.
27. K. L. NIELSEN, Methods in Numerical Analysis, MacMillan, New York, 1956.
28. S. J. RUUTH, Implicit-explicit Runge-Kutta methods for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), pp. 151-167.
29. B. I. SHRAIMAN, Order, disorder, and phase turbulence, Phys. Rev. Lett., 57 (1986), pp. 325-328.
30. G. D. SMITH, Numerical Solution of Partial Differential Equations, Oxford University Press, New York, 1965.
31. A. TAFLOVE, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, 1995.
32. L. N. TREFETHEN, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
33. L. N. TREFETHEN, 1-page, 1-second Matlab ETDRK4 code for Kuramoto-Sivashinsky equation, available online from http://web.comlab.ox. ac.uk/oucl/work/nick.trefethen/.
34. L. N. TREFETHEN, The PDE Coffee Table Book, in preparation.
35. J. R. QUAN AND C. T. CHANG, New insights in solving distributed system equations by the quadrature methods - I., Comput. Chem. Engrg., 13 (1989), pp. 779-788.
36. J. R. QUAN AND C. T. CHANG, New insights in solving distributed system equations by the quadrature methods - II., Comput. Chem. Engrg., 13 (1989), pp. 1017-1024.
37. J. A. C. WEIDEMAN AND S. C. REDDY, A MATLAB differentiation matrix suite, ACM Trans. Math. Software (TOMS), 26 (2000), pp. 465-519. (See also http:/dip.sun.ac.za/~weideman/research/differ.html.)
38. B. D. WELFERT, Generation of pseudospectral differentiation matrices I, SIAM J. Numer. Anal., 34 (1997), pp. 1640-1657.