Abstract
We present a linear rational pseudospectral (collocation) method with preassigned poles for solving boundary value problems. It consists in attaching poles to the trial polynomial so as to make it a rational interpolant. Its convergence is proved by transforming the problem into an associated boundary value problem. Numerical examples demonstrate that the rational pseudospectral method is often more efficient than the polynomial method.
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R. Baltensperger and J.-P. Berrut, The errors in calculating the pseudospectral differentiation matrices for Čebyšev–Gauss–Lobatto points, Comput. Math. Appl. 37 (1999) 41–48; Errata: 38 (1999) 119.
R. Baltensperger and J.-P. Berrut, The linear rational collocation method, J. Comput. Appl. Math. 134 (2001) 243–258.
C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, Singapore, 1978).
J.-P. Berrut, A pseudospectral Čebyšev method with preliminary transform to the circle: Ordinary differential equations, Report 232, Mathematisches Institut, Technische Universität München, Germany (1990); revised Université de Fribourg, Fribourg/Pérolles, Switzerland (1995).
J.-P. Berrut, The barycentric weights of rational interpolation with prescribed poles, J. Comput. Appl. Math. 86 (1997) 45–52.
J.-P. Berrut, Lagrange interpolation is better than its reputation, Report 01-2, Département de Mathématiques, Université de Fribourg, Suisse (July 2001).
J.-P. Berrut and R. Baltensperger, The linear rational collocation method for boundary value problems, BIT 41 (2001) 868–879.
J.-P. Berrut and H. Mittelmann, Rational interpolation through the optimal attachment of poles to the interpolating polynomial, Numer. Algorithms 23 (2000) 315–328.
J.-P. Berrut and H. Mittelmann, The linear rational pseudospectral method with iteratively optimized poles for two-point boundary value problems, SIAM J. Sci. Comput. 23 (2001) 961–975.
J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. revised (Dover, New York, 2001).
P. Canuto, M.Y. Hussaini, A. Quarteroni and T. Zang, Spectral Methods in Fluid Dynamics (Springer, New York, 1988).
H. Chen and B.D. Shizgal, A spectral solution of the Sturm–Liouville equation: Comparison of classical and nonclassical basis sets, J. Comput. Appl. Math. 136 (2001) 17–35.
C. Canuto and A. Quarteroni, Variational methods in the theoretical analysis of spectral approximations, in: Spectral Methods for Partial Differential Equations, eds. R.G. Voigt, D. Gottlieb and M.Y. Hussaini (SIAM, Philadelphia, PA, 1984) pp. 55–78.
B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge Univ. Press, Cambridge, 1996).
P. Henrici, Essentials of Numerical Analysis (Wiley, New York, 1982).
E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956).
F. Pérez-Acosta, L. Casasús and N. Hayek, Rational collocation for linear boundary value problems, J. Comput. Appl. Math. 33 (1990) 297–305.
H.R. Rutishauser, Vorlesungen über numerische Mathematik, Band 1 (Birkhäuser, Basel, 1976).
H.R. Schwarz, Numerische Mathematik, 4te Aufl. (Teubner, Stuttgart, 1997).
H.E. Salzer, Lagrangian interpolation at the Chebyshev points xn, v = cos(π/n), v = 0(1)n; some unnoted advantages, Computer J. 15 (1972) 156–159.
T. Tang and M.R. Trummer, Boundary layer resolving pseudospectral methods for singular perturbation problems, SIAM J. Sci. Comput. 17 (1996) 430–438.
G. Vainikko, The convergence of the collocation method for non-linear differential equations, USSR Comput. Math. Phys. 6 (1966) 47–58.
J.A.C. Weideman, Spectral methods based on nonclassical orthogonal polynomials, in: Approximations and Computation of Orthogonal Polynomials, eds. W. Gautschi, G. Golub and G. Opfer, International Series of Numerical Mathematics, Vol. 131 (Birkhäuser, Basel, 1999) pp. 239–251.
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Baltensperger, R., Berrut, JP. & Dubey, Y. The Linear Rational Pseudospectral Method with Preassigned Poles. Numerical Algorithms 33, 53–63 (2003). https://doi.org/10.1023/A:1025535231813
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DOI: https://doi.org/10.1023/A:1025535231813