Time-spectral solution of ordinary and partial differential equations is often regarded as an inefficient approach. The associated extension of the time domain, as compared to finite difference methods, is believed to result in uncomfortably many numerical operations and high memory requirements. It is shown in this work that performance is substantially enhanced by the introduction of algorithms for temporal and spatial subdomains in combination with sparse matrix methods. The accuracy and efficiency of the recently developed time spectral, generalized weighted residual method (GWRM) are compared to that of the explicit Lax-Wendroff and implicit Crank-Nicolson methods. Three initial-value PDEs are employed as model problems; the 1D Burger equation, a forced 1D wave equation and a coupled system of 14 linearized ideal magnetohydrodynamic (MHD) equations. It is found that the GWRM is more efficient than the time-stepping methods at high accuracies. The advantageous scalings <i>N<sub>t</sub></i><sup style="margin-left:-6px;">1.0</sup><i>N<sub>s</sub></i><sup style="margin-left:-6px;">1.43</sup> and <i>N<sub>t</sub></i><sup style="margin-left:-6px;">0.0</sup><i>N<sub>s</sub></i><sup style="margin-left:-6px;">1.08</sup> were obtained for CPU time and memory requirements, respectively, with <i> N t </i> and <i> N s </i> denoting the number of temporal and spatial subdomains. For time-averaged solution of the two-time-scales forced wave equation, GWRM performance exceeds that of the finite differenc e methods by an order of magnitude both in terms of CPU time and memory requirement. Favorable subdomain scaling is demonstrated for the MHD equations, indicating a potential for efficient solution of advanced initial-value problems in, for example, fluid mechanics and MHD.
Morchoisne, Y. (1979) Resolution of the Navier-Stokes Equations by a Space-Time Pseudospectral Method. La Recherche Aerospatiale, 5, 293-306.
Peyret, R. and Taylor, T.D. (1983) Computational Methods for Fluid Flow. Springer, New York. https://doi.org/10.1007/978-3-642-85952-6
Tal-Ezer, H. (1986) Spectral Methods in Time for Hyperbolic Equations. SIAM Journal on Numerical Analysis, 23, 11-26. https://doi.org/10.1137/0723002
Tal-Ezer, H. (1989) Spectral Methods in Time for Parabolic Problems. SIAM Journal on Numerical Analysis, 26, 1-11. https://doi.org/10.1137/0726001
Delic, G. (1987) Spectral Function Methods for Nonlinear Diffusion Equations. Journal of Mathematical Physics, 28, 39. https://doi.org/10.1063/1.527807
Dutt, P. (1990) Spectral Methods for Initial Boundary Value Problems: An Alternative Approach. SIAM Journal on Numerical Analysis, 27, 885-903. https://doi.org/10.1137/0727051
Ierley, B.S.G., Spencer, B. and Worthing, R. (1992) Spectral Methods in Time for a Class of Parabolic Partial Differential Equations. Journal of Computational Physics, 102, 88-97. https://doi.org/10.1016/S0021-9991(05)80008-7
Kosloff, D. and Tal-Ezer, H. (1993) A Modified Chebyshev Pseudospectral Method with an O(N -1 ) Time Step Restriction. Journal of Computational Physics, 104, 457-469. https://doi.org/10.1006/jcph.1993.1044
Zrahia, A.U. and Bar-Yoseph, P. (1994) Space-Time Spectral Element Method for Solution of Second-Order Hyperbolic Equations. Computer Methods in Applied Mechanics and Engineering, 116, 135-146. https://doi.org/10.1016/S0045-7825(94)80017-0
Bar-Yoseph, U.Z.P., Moses, E. and Yarin, A. (1995) Space-Time Spectral Element Methods for One-Dimensional Nonlinear Advection-Diffusion Problems. Journal of Computational Physics, 119, 62-74. https://doi.org/10.1006/jcph.1995.1116
Luo, Y. (1997) Polynomial Time-Marching for Three-Dimensional Wave Equations. Journal of Scientific Computing, 12, 465-477. https://doi.org/10.1023/A:1025633130781
Boyd, J.P. (2000) Chebyshev and Fourier Spectral Methods. Dover, New York.
Tang, J.-G. and Ma, H.-P. (2002) Single and Multi-Interval Legendre-Methods in Time for Parabolic Equations. Advances in Computational Mathematics, 17, 349-367. https://doi.org/10.1023/A:1016273820035
Lions, Y.M.J.-L. and Turinici, G. (2002) Resolution dEDP par un schema en temps parareel. C. R. Acad. Sci. Paris Sr. I Math., 7, 661.
Maerschalck, B.D. and Gerritsma, M.I. (2005) The Use of Chebyshev Polynomials in the Space-Time Least-Squares Spectral Element Method. Numerical Algorithms, 38, 173.
Gopinath, A.K. and Jameson, A. (2005) Time Spectral Method for Periodic Unsteady Computations over Two- and Three-Dimensional Bodies. 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, 10-13 January 2005, 14 p.
Canuto, A.Q.C., Hussaini, M.Y. and Tang, T.A. (2006) Spectral Methods, Fundamentals in Single Domains. Springer, Berlin.
Sicot, G.P.F. and Montagnac, M. (2008) Block-Jacobi Implicit Algorithms for the Time Spectral Method. AIAA Journal, 46, 3080-3089. https://doi.org/10.2514/1.36792
Dehghan, M. and Taleei, A. (2010) Numerical Solution of Nonlinear Schrödinger Equation by Using Time-Space Pseudo-Spectral Method. Numerical Methods for Partial Differential Equations, 26, 979-992.
Mavriplis, Z.Y.D.J., Yang, Z. and Mundis, N. (2012) Extensions of Time Spectral Methods for Practical Rotorcraft Problems. 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, 9-12 January 2012, Paper AIAA 2012-0423.
Emmett, M. and Minion, M. (2012) Toward an Efficient Parallel in Time Method for Partial Differential Equations. Communications in Applied Mathematics and Computational Science, 7, 105-132. https://doi.org/10.2140/camcos.2012.7.105
Luder, A.J. (2013) A Block-Jacobi Time-Spectral Method for Incompressible Flow. PhD Thesis, University of Michigan, Michigan.
Attar, P.J. (2015) Jacobian-Free Newton-Krylov Method for Implicit Time-Spectral Solution of the Compressible Navier-Stokes Equations. International Journal for Numerical Methods in Fluids, 79, 1-15. https://doi.org/10.1002/fld.4036
Scheffel, J. (2012) A Spectral Method in Time for Initial-Value Problems. American Journal of Computational Mathematics, 2, 173-193. https://doi.org/10.4236/ajcm.2012.23023
Scheffel, J. (2011) Time-Spectral Solution of Initial-Value Problems. In: Jang, C.-L., Ed., Partial Differential Equations: Theory, Analysis and Applications, Nova Science Publishers, Inc., Hauppauge, 1-49.
Fletcher, C.A.J. (2000) Computational Techniques for Fluid Dynamics. Springer, New York.
Dutt, A., Greengard, L. and Rokhlin, V. (2000) Spectral Deferred Correction Methods for Ordinary Differential Equations. BIT Numerical Mathematics, 40, 241-266. https://doi.org/10.1023/A:1022338906936
Gustafsson, B. and Kress, W. (2001) Deferred Correction Methods for Initial Value Problems. BIT Numerical Mathematics, 41, 986-995. https://doi.org/10.1023/A:1021937227950
Gustafsson, B. and Hemmingsson, L. (2002) Deferred Correction in Space and Time. Journal of Scientific Computing, 17, 541-550.
Huang, J., Jia, J. and Minion, M. (2006) Accelerating the Convergence of Spectral Deferred Correction Methods. Journal of Computational Physics, 214, 633-656. https://doi.org/10.1016/j.jcp.2005.10.004
Layton, A.T. (2009) On the Efficiency of Spectral Deferred Correction Methods for Time-Dependent Partial Differential Equations. Applied Numerical Mathematics, 59, 1629-1643. https://doi.org/10.1016/j.apnum.2008.11.004
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes. Cambridge University Press, Cambridge.
Mason, J.C. and Handscomb, D.C. (2003) Chebyshev Polynomials. Chapman and Hall, London.
Scheffel, J. and Håkansson, C. (2009) Solution of Systems of Nonlinear Equations, a Semi-Implicit Approach. Applied Numerical Mathematics, 59, 2430-2443. https://doi.org/10.1016/j.apnum.2009.05.002
Scheffel, J. and Mirza, A. (2012) Time-Spectral Solution of Initial-Value Problems-Subdomain Approach. American Journal of Computational Mathematics, 2, 72-81. https://doi.org/10.4236/ajcm.2012.22010
Scheffel, J., Lindvall, K. and Yik, H.F. (2018) A Time-Spectral Approach to Numerical Weather Prediction. Computer Physics Communications. https://doi.org/10.1016/j.cpc.2018.01.010
Bateman, G. (1978) MHD Instabilities. MIT Press, Cambridge.
Riva, F., Milanese, L. and Ricci, P. (2017) Uncertainty Propagation by Using Spectral Methods: A Practical Application to a Two-Dimensional Turbulence Fluid Model. Physics of Plasmas, 24, Article ID 102302. https://doi.org/10.1063/1.4996445