A Spectral Method in Time for Initial-Value Problems

DOI: 10.4236/ajcm.2012.23023   PDF   HTML   XML   6,755 Downloads   10,870 Views   Citations


A time-spectral method for solution of initial value partial differential equations is outlined. Multivariate Chebyshev series are used to represent all temporal, spatial and physical parameter domains in this generalized weighted residual method (GWRM). The approximate solutions obtained are thus analytical, finite order multivariate polynomials. The method avoids time step limitations. To determine the spectral coefficients, a system of algebraic equations is solved iteratively. A root solver, with excellent global convergence properties, has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by use of temporal and spatial subdomains. As examples of advanced application, stability problems within ideal and resistive magnetohydrodynamics (MHD) are solved. To introduce the method, solutions to a stiff ordinary differential equation are demonstrated and discussed. Subsequently, the GWRM is applied to the Burger and forced wave equations. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Thus the method shows potential for advanced initial value problems in fluid mechanics and MHD.

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J. Scheffel, "A Spectral Method in Time for Initial-Value Problems," American Journal of Computational Mathematics, Vol. 2 No. 3, 2012, pp. 173-193. doi: 10.4236/ajcm.2012.23023.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] R. D. Richtmyer and K. W. Morton, ”Difference Methods for Initial-value Problems”, Krieger Publishing, 1994.
[2] D. S. Harned and W. Kerner, ”Semi-implicit method for three-dimensional compressible magnetohydrodynamic simulation”, J. Comp. Phys. Vol. 60, No. 1, 1985, pp. 62-75. doi:10.1016/0021-9991(85)90017-8
[3] D. S. Harned and D. D. Schnack, ”Semi-implicit method for long time scale magnetohydrodynamic computations in three dimensions”, J. Comp. Phys. Vol. 65, No. 1, 1986, pp. 57-70. doi:10.1016/0021-9991(86)90004-5
[4] J. Scheffel, ”Semi-analytical solution of initial-value problems”, TRITA-ALF-2004-03, KTH Royal Institute of Technology, Stockholm, Sweden, 2004.
[5] B. A. Finlayson and L. E. Scriven, ”The Method of Weighted Residuals – a Review”, Appl. Mech. Rev. Vol. 19, No. 9, 1966, pp. 735-748.
[6] C. A. J. Fletcher, ”Computational Techniques for Fluid Dynamics”, Vols 1-2, Springer, 2000.
[7] R. Peyret and T. D. Taylor, ”Computational Methods for Fluid Flow”, Springer, New York 1983. doi:10.1007/978-3-642-85952-6
[8] H. Tal-Ezer, ”Spectral Methods in Time for Hyperbolic Equations”, SIAM J. Numer. Anal. Vol. 23, No. 1, 1986, pp. 11-26. doi:10.1137/0723002
[9] H. Tal-Ezer, ”Spectral Methods in Time for Parabolic Problems”, SIAM J. Numer. Anal. Vol. 26, No. 1, 1989, pp. 1-11. doi:10.1137/0726001
[10] G. Ierley, B. Spencer and R. Worthing, ”Spectral Methods in Time for a Class of Parabolic Partial Differential Equations”, J. Comput. Phys. Vol. 102, No. 1, 1992, pp. 88-97. doi:10.1016/S0021-9991(05)80008-7
[11] J.-G. Tang and H.-P. Ma, ”Single and Multi-interval Legendre ?-methods in Time for Parabolic Equations”, Adv. in Comput. Math. Vol. 17, No. 4, 2002, pp. 349-367. doi:10.1023/A:1016273820035
[12] M. Dehghan and A. Taleei, ”Numerical Solution of Nonlinear Schr?dinger Equation by Using Time-Space Pseudo-Spectral Method”, Num. Meth. for Part. Diff. Eq. Vol. 26, No. 4, 2010, pp. 979-992.
[13] D. Gottlieb and S. A. Orszag, ”Numerical Analysis of Spectral Methods: Theory and Applications”, SIAM, Philadelphia, 1987.
[14] J. C. Mason and D. C. Handscomb, ”Chebyshev Polynomials”, Chapman & Hall, 2003.
[15] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, ”Numerical Recipes”, Cambridge University Press, 1992.
[16] J. Scheffel and C. H?kansson, “Solution of Systems of Non-linear Equations-a Semi-implicit Approach”, Appl. Numer. Math. Vol. 59, No. 10, 2009,pp. 2430-2443. doi:10.1016/j.apnum.2009.05.002
[17] G. Dahlquist and ?. Bj?rck, ”Numerical Methods”, Prentice-Hall, 1974.
[18] A. M. Ostrowski, ”Solution of Equations and Systems of Equations”, Academic Press, New York and London, 1966.
[19] C. Moler, “Matlab News & Notes”, The MathWorks, Inc., May 2003.
[20] G. Bateman, “MHD Instabilities”, MIT Press, 1978.

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