zbMATH — the first resource for mathematics

Comparison of the discrete singular convolution algorithm and the Fourier pseudospectral method for solving partial differential equations. (English) Zbl 0993.65112
Summary: This paper explores the utility, tests the accuracy and examines the limitation of the discrete singular convolution (DSC) algorithm for solving partial differential equations (PDEs). The standard Fourier pseudospectral method is also implemented for a detailed comparison so that the performance of the DSC algorithm can be better evaluated. Three two-dimensional PDEs of different nature, the heat equation, the wave equation and the Navier-Stokes equation, are employed to make our assessment. Either the fourth-order Runge-Kutta or the Crank-Nicolson scheme is employed for the temporal discretization. The DSC algorithm is projected into the Fourier domain for analyzing its numerical resolution. It is demonstrated that the accuracy of the DSC algorithm is controllable. Comprehensive comparisons are given based on a variety of time increment, grid spacing, wave-number, and Reynolds number. It is found that the DSC algorithm is an accurate, stable and robust approach for solving these PDEs.

MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L05 Wave equation 35Q30 Navier-Stokes equations
Matlab
Full Text:
References:
 [1] Lanczos, C., Trigonometric interpolation of empirical and analytical functions, J. math. phys., 17, 123-199, (1938) · Zbl 0020.01301 [2] Cooley, J.W.; Tukey, J.W., An algorithm for the machine calculation of complex Fourier series, Math. comput., 19, 297-301, (1965) · Zbl 0127.09002 [3] Finlayson, B.A.; Scriven, L.E., The method of weighted residuals—a review, Appl. mech. rev., 19, 735-748, (1966) [4] Orszag, S.A., Comparison of pseudospectral and spectral approximations, Stud. appl. math., 51, 253-259, (1972) · Zbl 0282.65083 [5] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer Berlin · Zbl 0658.76001 [6] Fornberg, B., A practical guide to pseudospectral methods, (1996), Cambridge University Press Cambridge · Zbl 0844.65084 [7] Trefethen, L.N., Spectral methods in Matlab, (2000), Oxford University Oxford, England · Zbl 0953.68643 [8] Forsythe, G.E.; Wasow, W.R., Finite-difference methods for partial differential equations, (1960), Wiley New York · Zbl 0099.11103 [9] Isaacson, E.; Keller, H.B., Analysis of numerical methods, (1966), Wiley New York · Zbl 0168.13101 [10] Zienkiewicz, O.C., The finite element method in engineering science, (1971), McGraw-Hill London · Zbl 0237.73071 [11] Desai, C.S.; Abel, J.F., Introduction to the finite element methods, (1972), Van Nostrand Reinhold New York [12] Oden, J.T., The finite elements of nonlinear continua, (1972), McGraw-Hill New York · Zbl 0235.73038 [13] Nath, B., Fundamentals of finite elements for engineers, (1974), Athlone Press London [14] Fenner, R.T., Finite element methods for engineers, (1975), Imperial College Press London [15] Cheung, Y.K., Finite strip methods in structural analysis, (1976), Pergamon Press Oxford · Zbl 0375.73073 [16] Rao, S.S., The finite element method in engineering, (1982), Pergamon Press New York · Zbl 0472.73083 [17] Reddy, J.N., Energy and variational methods in applied mechanics, (1984), John Wiley New York · Zbl 0635.73017 [18] Wei, G.W., Discrete singular convolution for the fokker – planck equation, J. chem. phys., 110, 8930-8942, (1999) [19] Wei, G.W., A unified approach for solving the fokker – planck equation, J. phys. A, 33, 4935-4953, (2000) · Zbl 0988.82047 [20] Wei, G.W., Wavelets generated by using discrete singular convolution kernels, J. phys. A, 33, 8577-8596, (2000) · Zbl 0961.42019 [21] Wei, G.W., Solving quantum eigenvalue problems by discrete singular convolution, J. phys. B, 33, 343-352, (2000) [22] Wei, G.W., Vibration analysis by discrete singular convolution, J. sound vibration, 244, 535-553, (2001) · Zbl 1237.74095 [23] Wei, G.W., A new algorithm for solving some mechanical problems, Comput. methods appl. mech. engrg., 190, 2017-2030, (2001) · Zbl 1013.74081 [24] Wei, G.W., A unified method for computational mechanics, (), 1049-1054 · Zbl 1364.93338 [25] D.C. Wan, G.W. Wei, Discrete singular convolution-finite subdomain method for the solution of incompressible viscous flows, J. Comput. Phys., in press · Zbl 1130.76403 [26] Guan, S.; Lai, C.-H.; Wei, G.W., Bessel – fourier analysis of patterns in a circular domain, Physica D, 151, 83-98, (2001) · Zbl 1076.35535 [27] Wei, G.W., Synchronization of single-side averaged coupling and its application to shock capturing, Phys. rev. lett., 86, 3542-3545, (2001) [28] Wei, G.W., Discrete singular convolution method for the sine-Gordon equation, Physica D, 137, 247-259, (2000) · Zbl 0944.35087 [29] Wei, G.W., Discrete singular convolution for beam analysis, Engrg. structures, 23, 1045-1053, (2001) [30] Wei, G.W.; Zhao, Y.B.; Xiang, Y., The determination of the natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution, Int. J. mech. sci., 43, 1731-1746, (2001) · Zbl 1018.74017 [31] Y.B. Zhao, G.W. Wei, Y. Xiang, Plate vibration under irregular internal supports, Int. J. Solids Structures, in press · Zbl 1090.74603 [32] G.W. Wei, Y.B. Zhao, Y. Xiang, Discrete singular convolution and its application to the analysis of plates with internal supports. I. Theory and algorithm, Int. J. Numer. Methods Engrg., in press · Zbl 1058.74643 [33] Schwartz, L., Theore des distribution, (1951), Hermann Paris [34] Wei, G.W., Quasi wavelets and quasi interpolating wavelets, Chem. phys. lett., 296, 215-222, (1998) [35] Walter, G.G.; Blum, J., Probability density estimation using delta sequences, Ann. statist., 7, 328-340, (1979) · Zbl 0403.62025 [36] Finlayson, B.A., The method of weighted residuals and variational principles, (1972), Academic New York · Zbl 0319.49020 [37] Dow, J.O., A unified approach to the finite element method and error analysis procedures, (1999), Academic San Diego, CA · Zbl 0944.74001 [38] Sanders, B.F.; Katoposes, N.O.; Boyd, J.P., Spectral modeling of nonlinear dispersive waves, J. hydraulic. engrg. ASCE, 124, 2-12, (1998) [39] Kreiss, H.-O.; Oliger, J., Comparison of accurate methods for the integration of hyperbolic systems, Tellus, 24, 199-215, (1972) [40] Roberts, K.V.; Weiss, N.O., Math. comput., 20, 272, (1966) [41] Swartz, B.; Wendroff, B., The relation between the Galerkin and collocation methods using smooth splines, SIAM J. numer. anal., 11, 994, (1974) · Zbl 0288.65066 [42] Vichnevetsky, R.; Bowles, J.B., Fourier analysis of numerical approximations of hyperbolic equations, (1982), SIAM Philadelphia · Zbl 0495.65041 [43] Yang, H.H.; Shizgal, B., Chebyshev pseudospectral multi-domain technique for viscous flow calculation, Comput. methods appl. mech. engrg., 118, 47, (1994) · Zbl 0848.76071 [44] Le Quéré, P.; Alziary de Roquefort, T., Computation of natural convection in two-dimension cavities, J. comput. phys., 57, 210, (1985) · Zbl 0585.76128
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.