Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-29T08:09:09.166Z Has data issue: false hasContentIssue false
  • English
  • Français

A review of pseudospectral methods for solving partial differential equations

Published online by Cambridge University Press:  07 November 2008

Bengt Fornberg  and
David M. Sloan
Bengt Fornberg
Affiliation:
Corporate ResearchExxon Research and Engineering Company Annandale, NJ 08801, USA E-mail: bfornbe@erenj.com
David M. Sloan
Affiliation:
Department of MathematicsUniversity of StrathclydeGlasgow G1 1XH, Scotland E-mail: caas10@computer-centre-sun.strathclyde.ac.uk
Get access Rights & Permissions [Opens in a new window]

Extract

Finite Difference (FD) methods approximate derivatives of a function by local arguments (such as du(x) / dx ≈ (u(x + h) − u(x − h))/2h, where h is a small grid spacing) – these methods are typically designed to be exact for polynomials of low orders. This approach is very reasonable: since the derivative is a local property of a function, it makes little sense (and is costly) to invoke many function values far away from the point of interest.

Type
Research Article
Information
Acta Numerica , Volume 3 , January 1994 , pp. 203 - 267
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abarbanel, S., Gottlieb, D. and Tadmor, E. (1986), ‘Spectral methods for discontinuous problems’, in Numerical Methods for Fluid Dynamics II (Morton, K.W. and Baines, M.J., eds), Clarendon Press (Oxford), 129–153.Google Scholar
Bayliss, A., Gottlieb, D., Matkowsky, B.J. and Minkoff, M. (1989), ‘An adaptive pseudospectral method for reaction diffusion problems’, J. Comput. Phys. 81, 421–443.CrossRefGoogle Scholar
Bayliss, A. and Turkel, E. (1992), ‘Mappings and accuracy for Chebyshev pseudospectral approximations’, J. Comput. Phys. 101, 349–359.CrossRefGoogle Scholar
Beylkin, G., Coifman, R. and Rokhlin, V. (1991), ‘Fast wavelet transforms and numerical algorithms’, Comm. Pure Appl. Math. 44, 141–183.CrossRefGoogle Scholar
Boyd, J.P. (1989), Chebyshev and Fourier Spectral Methods, Springer (New York)CrossRefGoogle Scholar
Boyd, J.P. (1992), ‘Multipole expansions and pseudospectral cardinal functions: a new generalization of the fast Fourier transform’, J. Comput. Phys. 103, 184–186.CrossRefGoogle Scholar
Breuer, K.S. and Everson, R.M. (1992), ‘On the errors incurred calculating derivatives using Chebyshev polynomials’, J. Comput. Phys. 99, 56–67.CrossRefGoogle Scholar
Cai, W., Gottlieb, D. and Shu, C.-W. (1989), ‘Essentially nonoscillatory spectral Fourier methods for shock wave calculation’, Math. Comput. 52, 389–410.Google Scholar
Cai, W., Gottlieb, D. and Shu, C.W. (1992), ‘One-sided filters for spectral Fourier approximation of discontinuous functions’, SIAM J. Numer. Anal. 29, 905–916.CrossRefGoogle Scholar
Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T. (1988), Spectral Methods in Fluid Dynamics, Springer (New York).CrossRefGoogle Scholar
Canuto, C. and Pietra, P. (1987), ‘Boundary and interface conditions with a FE pre-conditioner for spectral methods’. Report No. 553, I.A.N., Pavia University, Italy.Google Scholar
Canuto, C. and Quarteroni, A. (1985), ‘Preconditioned minimal residual methods for Chebyshev spectral calculations’, J. Comput. Phys. 60, 315–337.CrossRefGoogle Scholar
Canuto, C. and Quarteroni, A. (1987), ‘On the boundary treatment in spectral method for hyperbolic systems’, J. Comput. Phys. 71, 100–110.CrossRefGoogle Scholar
Cheney, E.W. (1966), Introduction to Approximation Theory McGraw-Hill (New York).Google Scholar
Cooley, J.W. and Tukey, J.W. (1965), ‘An algorithm for the machine calculation of complex Fourier series’, Math. Comput. 19, 297–301.CrossRefGoogle Scholar
Dahlquist, G. (1956), ‘Convergence and stability in the numerical integration of ordinary differential equations’, Math. Scand. 4, 33–53.CrossRefGoogle Scholar
Dahlquist, G. (1985), ‘33 years of numerical instability, part I’, BIT 25, 188–204.CrossRefGoogle Scholar
Davis, P.J. (1975), Interpolation and Approximation, Dover (New York).Google Scholar
Davis, P.J. and Rabinowitz, P. (1984), Methods of Numerical Integration, 2nd Edn, Academic Press (London, New York).Google Scholar
Deville, M. and Mund, E. (1985), ‘Chebyshev PS solution of second-order elliptic equations with finite element preconditioning’, J. Comput. Phys. 60, 517–553.CrossRefGoogle Scholar
Dubiner, M. (1987), ‘Asymptotic analysis of spectral methods’, J. Sci. Comput. 2, 3–31.CrossRefGoogle Scholar
Eisen, H. and Heinrichs, W. (1992), ‘A new method of stabilization for singular perturbation problems with spectral methods’, SIAM J. Numer. Anal. 29, 107–122.CrossRefGoogle Scholar
Fornberg, B. (1975), ‘On a Fourier method for the integration of hyperbolic equations’, SIAM J. Numer. Anal. 12, 509–528.CrossRefGoogle Scholar
Fornberg, B. (1987), ‘The pseudospectral method: comparisons with finite differences for the elastic wave equation’, Geophysics 52, 483–501.CrossRefGoogle Scholar
Fornberg, B. (1988a), ‘The pseudospectral method: accurate representation of interfaces in elastic wave calculations’, Geophysics 53, 625–637.CrossRefGoogle Scholar
Fornberg, B. (1988b), ‘Generation of finite difference formulae on arbitrarily spaced grids’, Math. Comput. 51, 699–706.CrossRefGoogle Scholar
Fornberg, B. (1990a), ‘High order finite differences and the pseudospectral method on staggered grids’, SIAM J. Num. Anal. 27, 904–918.CrossRefGoogle Scholar
Fornberg, B. (1990b), ‘An improved pseudospectral method for initial-boundary value problems’, J. Comput. Phys. 91, 381–397.CrossRefGoogle Scholar
Fornberg, B. (1992), ‘Fast generation of weights in finite difference formulas’, in Recent Developments in Numerical Methods and Software for ODEs/DAEs/PDEs (Byrne, G.D. and Schiesser, W.E., eds), World Scientific (Singapore), 97–123.CrossRefGoogle Scholar
Fornberg, B. (1994), ‘A pseudospectral approach for polar and spherical geometries’, SIAM J. Sci. Comput. submitted.Google Scholar
Fornberg, B. and Whitham, G.B. (1978), ‘A numerical and theoretical study of certain nonlinear wave phenomena’, Phil. Trans. Roy. Soc. London A 289, 373–404.Google Scholar
Fourier, J.B.J. (1822), Théorie analytique de la chaleur, Paris.Google Scholar
Funaro, D. (1987), ‘A preconditioned matrix for the Chebyshev differencing operator’, SIAM J. Numer. Anal. 24, 1024–1031.CrossRefGoogle Scholar
Funaro, D. (1992), Polynomial Approximation of Differential Equations Lecture Notes in Physics 8, Springer (Berlin).CrossRefGoogle Scholar
Funaro, D. and Gottlieb, D. (1988), ‘A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations’, Math. Comput. 51, 599–613.CrossRefGoogle Scholar
Gaier, D. (1987), Lectures on Complex Approximation, Birkhäuser (Boston).CrossRefGoogle Scholar
Gear, C.W. (1971), Numerical Solution of Ordinary and Partial Differential Equations, Prentice Hall (Englewood Cliffs, NJ).Google Scholar
Goldberg, M. and Tadmor, E. (1985), ‘Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems’, Math. Comput. 44, 361–377.CrossRefGoogle Scholar
Gottlieb, D. and Lustman, L. (1983), ‘The spectrum of the Chebyshev collocation operator for the heat equation’, SIAM J. Numer. Anal. 20, 909–921.CrossRefGoogle Scholar
Gottlieb, D., Lustman, L. and Tadmor, E. (1987), ‘Stability analysis of spectral methods for hyperbolic initial-boundary value problems’, SIAM J. Num. Anal. 24, 241–258.CrossRefGoogle Scholar
Gottlieb, D. and Orszag, S.A. (1977), Numerical Analysis of Spectral Methods, SIAM (Philadelphia).CrossRefGoogle Scholar
Gottlieb, D. and Tadmor, E. (1991), ‘The CFL condition for spectral approximation to hyperbolic BVPs’, Math. Comput. 56, 565–588.CrossRefGoogle Scholar
Gottlieb, D. and Turkel, E. (1980), ‘On time discretization for spectral methods’, Stud. Appl. Math. 63, 67–86.CrossRefGoogle Scholar
Greshko, P.M. and Lee, R.L. (1981), ‘Don't suppress the wiggles – they're trying to tell you something’, Computers and Fluids 9, 223–253.CrossRefGoogle Scholar
Gustafsson, B., Kreiss, H.-O. and Sundstrom, A. (1972), ‘Stabilitytheory of difference approximations for mixed initial-boundary value problems II’, Math. Comput. 26, 649–685.CrossRefGoogle Scholar
Hairer, E., Nørsett, S.P. and Wanner, G. (1987), Solving Ordinary Differential Equations I – Nonstiff Problems Springer (Berlin).CrossRefGoogle Scholar
Hairer, E. and Wanner, G. (1991), Solving Ordinary Differential Equations II – Stiff and Differential-Algebraic Problems, Springer (Berlin).Google Scholar
Haldenwang, P., Labrosse, G., Abboudi, S. and Deville, M. (1984), ‘Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation’, J. Comput. Phys. 55, 115–128.CrossRefGoogle Scholar
Holberg, O. (1987), ‘Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena’, Geophys. Prospecting 35, 629–655.CrossRefGoogle Scholar
Hou, Y.-C. C. and Kreiss, H.-O. (1993), ‘Comparison of finite difference and the pseudo-spectral approximations for hyperbolic equations’, to be published.Google Scholar
Huang, W. and Sloan, D.M. (1993a), ‘Pole condition for singular problems: the pseudospectral approximation’, J. Comput. Phys. 107, 254–261.CrossRefGoogle Scholar
Huang, W. and Sloan, D.M. (1993b), ‘A new pseudospectral method with upwind features’, IMA. J. Numer. Anal. 13, 413–430.CrossRefGoogle Scholar
Huang, W. and Sloan, D.M. (1993c), ‘The pseudospectral method for solving differential eigenvalue problems’, J. Comput. Phys. to appear.Google Scholar
Hussaini, M.Y. and Zang, T.A. (1984), ‘Iterative spectral methods and spectral solution to compressible flows’, in Spectral Methods for PDEs (Voigt, R., Gottlieb, D. and Hussaini, M.Y., eds), SIAM (Philadelphia).Google Scholar
Karageorghis, A. (1988), ‘A note on the Chebyshev coefficients of the general order derivative of an infinitely differentiable function’, J. Comput. Appl. Math. 21, 129–132.CrossRefGoogle Scholar
Kindelan, M., Kamel, A. and Sguazzero, P. (1990), ‘On the construction and efficiency of staggered numerical differentiators for the wave equation’, Geophysics 55, 107–110.CrossRefGoogle Scholar
Kosloff, D. and Tal-Ezer, H. (1993), ‘A modified Chebyshev pseudospectral method with an (N −1) time step restriction’, J. Comput. Phys. 104, 457–469.CrossRefGoogle Scholar
Kreiss, H.-O. (1962), ‘Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungenapproximieren’, Nordisk Tidskr. Informationsbehandling 2, 153–181.Google Scholar
Kreiss, H.-O. (1968), ‘Stability theory for difference approximations of mixed initial boundary value problems I’, Math. Comput. 22, 703–714.CrossRefGoogle Scholar
Kreiss, H.-O. (1970), ‘Initial boundary value problems for hyperbolic systems’, Comm. Pure Appl. Math. 23, 277–288.CrossRefGoogle Scholar
Kreiss, H.-O. and Oliger, J. (1972), ‘Comparison of accurate methods for the integration of hyperbolic equations’, Tellus XXIV, 199–215.CrossRefGoogle Scholar
Krylov, V.I. (1962), Approximate Calculation of Integrals, Macmillan (New York).Google Scholar
Lambert, J.D. (1991), Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, Wiley (Chichester, UK).Google Scholar
Lanczos, C. (1938), ‘Trigonometric interpolation of empirical and analytical functions’, J. Math. Phys. 17, 123–199.CrossRefGoogle Scholar
Lele, S.K. (1992), ‘Compact finite difference schemes with spectral-like resolution’, J. Comput. Phys. 103, 16–42.CrossRefGoogle Scholar
Maday, Y., Kaber, S.M.O. and Tadmor, E. (1993), ‘Legendre PS viscosity methods for nonlinear conservation laws’, SIAM J. Numer. Anal. 30, 321–342.CrossRefGoogle Scholar
Majda, A., McDonough, J. and Osher, S. (1978), ‘The Fourier method for non-smooth initial data’, Math. Comput. 32, 1041–1081.CrossRefGoogle Scholar
Markushevich, A.I. (1967), Theory of Functions of a Complex Variable, Vol. III (Transl. by Silverman, R.A.), Prentice Hall (New York).Google Scholar
Mercier, B. (1989), An Introduction to the Numerical Analysis of Spectral Methods, Springer (Berlin).Google Scholar
Merryfield, W.J. and Shizgal, B. (1993), ‘Properties of collocation third-derivative operators’, J. Comput. Phys. 105, 182–185.CrossRefGoogle Scholar
Mittet, R., Holberg, O., Arntsen, B. and Amundsen, L. (1988), ‘Fastfinite difference modeling of 3-D elastic wave equation’, Society of Exploration Geophysics Expanded Abstracts I, 1308–1311.Google Scholar
Mulholland, L.S. and Sloan, D.M. (1992), ‘The role of preconditioning in the solution of evolutionary PDEs by implicit Fourier PS methods’, J. Comput. Appl. Math. 42, 157–174.CrossRefGoogle Scholar
Nielsen, K.L. (1956), Methods in Numerical Analysis, Macmillan (New York).Google Scholar
Orszag, S.A. (1969), ‘Numerical methods for the simulation of turbulence’, Phys. Fluids Suppl. II 12, 250–257.CrossRefGoogle Scholar
Orszag, S.A. (1970), ‘Transform method for calculation of vector coupled sums: Application to the spectral form of the vorticity equation’, J. Atmos. Sci. 27, 890–895.2.0.CO;2>CrossRefGoogle Scholar
Orszag, S.A. (1972), ‘Comparison of pseudospectral and spectral approximations’, Stud. Appl. Math. 51, 253–259.CrossRefGoogle Scholar
Phillips, T.N. (1988), ‘On the Legendre coefficients of a general-order derivative of an infinitely differentiable function’, IMA J. Numer. Anal. 8, 455–459.CrossRefGoogle Scholar
Phillips, T.N., Zang, T.A. and Hussaini, M.Y. (1986), ‘Preconditioners for the spectral multigrid method’, IMA J. Numer. Anal. 6, 273–292.CrossRefGoogle Scholar
Powell, M.J.D. (1981), Approximation Theory and Methods, Cambridge University Press (Cambridge).CrossRefGoogle Scholar
Reddy, S.C. and Trefethen, L.N. (1990), ‘Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues’, Comput. Meth. Appl. Mech. Engrg 80, 147–164.CrossRefGoogle Scholar
Richtmyer, R.D. and Morton, K.W. (1967), Difference Methods for Initial-value Problems, 2nd Edn, Wiley (New York).Google Scholar
Rivlin, T.J. (1969), An Introduction to the Approximation of Functions, Dover (New York).Google Scholar
Sand, J. and Østerby, O. (1979), ‘Regions of absolute stability’, Report DAIMI PP-102, Computer Science Department, Aarhus University, Denmark.CrossRefGoogle Scholar
Sansone, G. (1959), Orthogonal Functions, Interscience (New York).Google Scholar
Shampine, L.F. and Gordon, M.K. (1975), Computer Solution of Ordinary Differential Equations, Freeman, W.H. (San Francisco).Google Scholar
Solomonoff, A. (1992), ‘A fast algorithm for spectral differentiation’, J. Comput. Phys. 98, 174–177.CrossRefGoogle Scholar
Solomonoff, A. (1994), ‘Bayes finite difference schemes’, SIAM J. Num. Anal. submitted.Google Scholar
Solomonoff, A. and Turkel, E. (1989), ‘Global properties of pseudospectral methods’, J. Comput. Phys. 81, 239–276.CrossRefGoogle Scholar
Stenger, F. (1981), ‘Numerical methods based on Whittaker cardinal, or sinc functions’, SIAM Review 23, 165–224.CrossRefGoogle Scholar
Szegő, G. (1959), Orthogonal Polynomials, American Mathematical Society (Washington, D.C.).Google Scholar
Tadmor, E. (1987), ‘Stability analysis of finite-difference, pseudospectral and Fourier–Galerkin approximations for time-dependent problems’, SIAM Review 29, 525–555.CrossRefGoogle Scholar
Tadmor, E. (1989), ‘Convergence of spectral methods for nonlinear conservation laws’, SIAM J. Num. Anal. 26, 30–44.CrossRefGoogle Scholar
Tadmor, E. (1990), ‘Shock capturing by the spectral viscosity method’, Comput. Meth. Appl. Mech. Engrg 80, 197–208.CrossRefGoogle Scholar
Tadmor, E. (1993), ‘Superviscosity and spectral approximations of nonlinear conservation laws’, in Numerical Methods for Fluid Dynamics IV (Baines, M.J. and Morton, K.W., eds), Oxford University Press (Oxford), 69–82.Google Scholar
Taylor, T.D., Hirsh, R.S. and Nadworny, N.M. (1984), ‘Comparison of FFT, direct inversion and conjugate gradient methods for use in pseudospectral methods’, Comput. Fluids 12, 1–9.CrossRefGoogle Scholar
Trefethen, L.N. (1983), ‘Group velocity interpretation of the stability theory of Gustafsson, Kreiss and Sundstrom’, J. Comput. Phys. 49, 199–217.CrossRefGoogle Scholar
Trefethen, L.N. (1988), ‘Lax-stability vs. eigenvalue stability of spectral methods’, in Numerical Methods for Fluid Dynamics III (Morton, K.W. and Baines, M.J., eds), Clarendon Press (Oxford), 237–253.Google Scholar
Trefethen, L.N. and Trummer, M.R. (1987), ‘An instability phenomenon in spectral methods’, SIAM J. Numer. Anal. 24, 1008–1023.CrossRefGoogle Scholar
Trefethen, L.N. and Weideman, J.A.C. (1991), ‘Two results on polynomial interpolation in equally spaced points’, J. Approx. Theory 65, 247–260.CrossRefGoogle Scholar
Voigt, R.G., Gottlieb, D. and Hussani, M.Y. (eds) (1984), Spectral Methods for Partial Differential Equations SIAM (Philadelphia).Google Scholar
Walsh, J.L. (1960), Interpolation and Approximation by Rational Functions in the Complex Domain, Colloquium Publications of the Amer. Math. Soc, 20, 3rd Edn.Google Scholar
Weideman, J.A.C. (1992), ‘The eigenvalues of Hermite and rational spectral DMs’, Numer. Math. 61, 409–432.CrossRefGoogle Scholar
Weideman, J.A.C. and Trefethen, L.N. (1988), ‘The eigenvalues of second-order spectral differentiation matrices’, SIAM J. Numer. Anal. 25, 1279–1298.CrossRefGoogle Scholar
Welfert, B.D. (1993), ‘A remark on pseudospectral differentiation matrices’, submitted to SIAM J. Num. Anal.Google Scholar
Whittaker, E.T. (1915), ‘On the functions which are represented by the expansions of the interpolation theory’, Proc. Roy. Soc. Edinburgh 35, 181–194.Google Scholar
Whittaker, J.M. (1927), ‘On the cardinal function of interpolation theory’, Proc. Edinburgh Math. Soc. Ser. 1 2, 41–46.CrossRefGoogle Scholar