Abstract
We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use local polynomial approximations to correct the solution. We analyze the accuracy and eigenvalue stability of the method for several PDEs. The method compares favorably to traditional spectral methods, and numerical results indicate that for hyperbolic problems a time step restriction of O(1/N) is sufficient for stability.
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Aydin, O., Ünal, A., Ayhan, T.: Natural convection in rectangular enclosures heated from one side and cooled from the ceiling. Int. J. Heat Mass Transfer 42(13), 2345–2355 (1999)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, Mineola (2001)
Boyd, J.P.: Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithms. J. Comput. Phys. 199(2), 688–716 (2004)
Boyd, J.P.: Fourier embedded domain methods: extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and C ∞. Appl. Math. Comput. 161(2), 591–597 (2005)
Boyd, J.P.: Asymptotic Fourier coefficients for a C ∞ bell (smoothed-“top-hat”) & the Fourier extension problem. J. Sci. Comput. 29(1), 1–24 (2006)
Boyd, J.P.: Exponentially accurate Runge–Free approximation of non-periodic functions from samples on an evenly–spaced grid. Appl. Math. Lett. 20(9), 971–975 (2007)
Boyd, J.P., Ong, J.R.: Exponentially–convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, part I: single-interval schemes. Commun. Comput. Phys. 5(2–4), 484–497 (2009)
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)
Farhangnia, M., Biringen, S., Peltier, L.J.: Numerical simulation of two-dimensional buoyancy-driven turbulence in a tall rectangular cavity. Int. J. Numer. Methods Fluids 23(12), 1311–1326 (1996)
Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)
Gelb, A.: Reconstruction of piecewise smooth functions from non-uniform grid point data. J. Sci. Comput. 30(3), 409–440 (2007)
Gelb, A., Platte, R.B., Rosenthal, W.S.: The discrete orthogonal polynomial least squares method for approximation and solving partial differential equations. Commun. Comput. Phys. 3(3), 734–758 (2008)
Gelb, A., Tanner, J.: Robust reprojection methods for the resolution of the Gibbs phenomenon. Appl. Comput. Harmon. Anal. 20(1), 3–25 (2006)
Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)
Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995)
Hale, N., Trefethen, L.N.: New quadrature methods from conformal maps. SIAM J. Numer. Anal. 46(2), 930–948 (2008)
Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)
Kosloff, D., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an O(N −1) time step restriction. J. Comput. Phys. 104(2), 457–469 (1993)
Platte, R.B., Driscoll, T.A.: Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM J. Numer. Anal. 43(2), 750–766 (2005)
Reddy, S.C., Trefethen, L.N.: Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues. Comput. Methods Appl. Mech. Eng. 80(1-3), 147–164 (1990)
Sarris, I.E., Lekakis, I., Vlachos, N.S.: Natural convection in rectangular tanks heated locally from below. Int. J. Heat Mass Transfer 47(14-16), 3549–3563 (2004)
Tadmor, E.: The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23(1), 1–10 (1986)
Tadmor, E., Tanner, J.: Adaptive filters for piecewise smooth spectral data. IMA J. Numer. Anal. 25(4), 635–647 (2005)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)
Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50(1), 67–87 (2008)
Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. Princeton University Press, Princeton (2005)
Weideman, J.A.C., Trefethen, L.N.: The eigenvalues of second-order spectral differentiation matrices. SIAM J. Numer. Anal. 25(6), 1279–1298 (1988)
Xia, C., Murthy, J.Y.: Buoyancy–driven flow transitions in deep cavities heated from below. J. Heat Transfer 124(4), 650–659 (2002)
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R.B. Platte’s address after December 2009: Arizona State University, Department of Mathematics and Statistics, Tempe, AZ, 85287-1804.
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Platte, R.B., Gelb, A. A Hybrid Fourier–Chebyshev Method for Partial Differential Equations. J Sci Comput 39, 244–264 (2009). https://doi.org/10.1007/s10915-008-9264-y
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DOI: https://doi.org/10.1007/s10915-008-9264-y