330
Views
9
CrossRef citations to date
0
Altmetric
Section B

Legendre–Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations

&
Pages 1434-1460 | Received 17 Apr 2013, Accepted 01 Sep 2013, Published online: 29 Oct 2013

References

  • O. Axelsson, A class of A-stable methods, BIT 9 (1969), pp. 185–199. doi: 10.1007/BF01946812
  • I. Babuška and T. Janik, The h-p version of the finite element method for parabolic equations: I. The p-version in time, Numer. Methods Partial Differ. Equ. 5 (1989), pp. 363–399. doi: 10.1002/num.1690050407
  • I. Babuška and T. Janik, The h-p version of the finite element method for parabolic equations: II. The h-p version in time, Numer. Methods Partial Differ. Equ. 6 (1990), pp. 343–369. doi: 10.1002/num.1690060406
  • P. Bar-Yoseph, E. Moses, U. Zrahia, and A.L. Yarin, Space-time spectral element methods for one-dimensional nonlinear advection-diffusion problems, J. Comput. Phys. 119 (1995), pp. 62–74. doi: 10.1006/jcph.1995.1116
  • A. Bayliss, A. Class, and B. Matkowsky, Roundoff error in computing derivatives using the Chebyshev differentiation matrix, J. Comput. Phys. 116 (1994), pp. 380–383. doi: 10.1006/jcph.1995.1036
  • C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis, Vol. 5, P.G. Ciarlet and J.L. Lions, eds., North-Holland, Amsterdam, 1997.
  • J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Publications, New York, 2001.
  • J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations, Runge–Kutta and General Linear Methods, Wiley and Sons, Chichester, 1987.
  • C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
  • F.H. Chipman, A-stable Runge–Kutta processes, BIT 11 (1971), pp. 384–388. doi: 10.1007/BF01939406
  • A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2000), pp. 241–266. doi: 10.1023/A:1022338906936
  • D. Funaro, Polynomial Approximations of Differential Equations, Springer-Verlag, Berlin, 1992.
  • I. Glenn, S. Brian, and W. Rodney, Spectral methods in time for a class of parabolic partial differential equations, J. Comput. Phys. 102 (1992), pp. 88–97. doi: 10.1016/S0021-9991(05)80008-7
  • B.-Y. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998.
  • B.-Y. Guo and Z-Q. Wang, Legendre–Gauss collocation methods for ordinary differential equations, Adv. Comput. Math. 30 (2009), pp. 249–280. doi: 10.1007/s10444-008-9067-6
  • B.-Y. Guo and Z-Q. Wang, A spectral collocation method for solving initial value problems of first order ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), pp. 1029–1054. doi: 10.3934/dcdsb.2010.14.1029
  • E. Hairer, S.P. Norsett, and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems, Springer-Verlag, Berlin, 1987.
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equation II: Stiff and Differential–Algebraic Problems, Springer-Verlag, Berlin, 1991.
  • L.V. Kantorovich, On a new method of approximate solution of partial differential equation, Dokl. Akad. Nauk SSSR 4 (1934), pp. 532–536.
  • N. Kanyamee and Z-M. Zhang, Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems, Int. J. Numer. Anal. Model. 8 (2011), pp. 86–104.
  • H.O. Kreiss and J. Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 (1972), pp. 199–215. doi: 10.1111/j.2153-3490.1972.tb01547.x
  • J.D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem, Wiley and Sons, Chichester, 1991.
  • R.K. Mohanty, M.K. Jain and K. George, On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients, J. Comput. Appl. Math. 72 (1996), pp. 421–431. doi: 10.1016/0377-0427(96)00011-8
  • S.A. Orszag, Comparison of pseudospectral and spectral approximations, Stud. Appl. Math. 51 (1972), pp. 253–259.
  • A. Prothero and A. Robinson, On the stability and the accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comput. 28 (1974), pp. 145–162. doi: 10.1090/S0025-5718-1974-0331793-2
  • D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal. 38 (2000), pp. 837–875. doi: 10.1137/S0036142999352394
  • D. Schötzau and C. Schwab, An hp a-priori error analysis of the DG time-stepping method for initial value problems, Calcolo 37 (2000), pp. 207–232. doi: 10.1007/s100920070002
  • J. Shen and L.-L. Wang, Fourierization of the Legendre–Galerkin method and a new space-time spectral method, Appl. Numer. Math. 57 (2007), pp. 710–720. doi: 10.1016/j.apnum.2006.07.012
  • J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer, Heidelberg, 2011.
  • J.C. Slater, Electronic energy bands in metal, Phys. Rev. 45 (1934), pp. 794–801. doi: 10.1103/PhysRev.45.794
  • A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.
  • H. Tal-Ezer, Spectral methods in time for hyperbolic equations, SIAM J. Numer. Anal. 23 (1986), pp. 11–26. doi: 10.1137/0723002
  • H. Tal-Ezer, Spectral methods in time for parabolic problems, SIAM J. Numer. Anal. 26 (1989), pp. 1–11. doi: 10.1137/0726001
  • J.-G. Tang and H.-P. Ma, Single and multi-interval Legendre τ-methods in time for parabolic equations, Adv. Comput. Math. 17 (2002), pp. 349–367. doi: 10.1023/A:1016273820035
  • J.-G. Tang and H.-P. Ma, A Legendre spectral method in time for first-order hyperbolic equations, Appl. Numer. Math. 57 (2007), pp. 1–11. doi: 10.1016/j.apnum.2005.11.009
  • T. Tang and X. Xu Accuracy enhancement using spectral postprocessing for differential equations and integral equations, Commun. Comput. Phys. 5 (2009), pp. 779–792.
  • Z.-Q. Wang and B.-Y. Guo, Legendre–Gauss–Radau collocation method for solving initial value problems of first order ordinary differential equations, J. Sci. Comput. 52 (2012), pp. 226–255. doi: 10.1007/s10915-011-9538-7
  • T.P. Wihler, An a priori error analysis of the hp-version of the continuous Galerkin FEM for nonlinear initial value problems, J. Sci. Comput. 25 (2005), pp. 523–549. doi: 10.1007/s10915-004-4796-2
  • K. Wright, Some relationship between implicit Runge–Kutta, collocation and τ-methods and their stability properties, BIT 10 (1970), pp. 217–227. doi: 10.1007/BF01936868
  • U. Zrahia and P. Bar-Yoseph, Space-time spectral element method for solution of second-order hyperbolic equations, ICOSAHOM’92 (Montpellier, 1992), Comput. Methods Appl. Mech. Eng. 116 (1994), pp. 135–146. doi: 10.1016/S0045-7825(94)80017-0

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.