A MATLAB differentiation matrix suite

A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

[1]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[2]  V. Rokhlin,et al.  Fast algorithms for polynomial interpolation, integration, and differentiation , 1996 .

[3]  K. S. Kölbig,et al.  Errata: Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C., 1994, and all known reprints , 1972 .

[4]  J. L. Schonfelder,et al.  Chebyshev expansions for the error and related functions , 1978 .

[5]  Dan S. Henningson,et al.  On stability of streamwise streaks and transition thresholds in plane channel flows , 1998, Journal of Fluid Mechanics.

[6]  A. Bayliss,et al.  Roundoff Error in Computing Derivatives Using the Chebyshev Differentiation Matrix , 1995 .

[7]  Peter Henrici,et al.  Discrete Fourier analysis, Cauchy integrals, construction of conformal maps, univalent functions , 1986 .

[8]  J. Pryce.  Numerical Solution of Sturm-Liouville Problems , 1994 .

[9]  Ambikeshwar Sharma.  Some Poised and Nonpoised Problems of Interpolation , 1972 .

[10]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[11]  John Lund,et al.  Sinc function computation of the eigenvalues of Sturm-Liouville problems , 1987 .

[12]  Jet Wimp,et al.  Computation with recurrence relations , 1986 .

[13]  Tao Tang,et al.  Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems , 1996, SIAM J. Sci. Comput..

[14]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[15]  F. Stenger.  Numerical Methods Based on Sinc and Analytic Functions , 1993 .

[16]  S. Flügge,et al.  Practical Quantum Mechanics , 1976 .

[17]  L. Trefethen,et al.  THE EIGENVALUES OF SECOND-ORDER SPECTRAL DIFFERENTIATION MATRICES* , 1988 .

[18]  S. Orszag,et al.  Theory and applications of spectral methods , 1984 .

[19]  Jean-Paul Berrut,et al.  Barycentric formulae for cardinal (SINC-)interpolants , 1989 .

[20]  Leslie Greengard,et al.  Spectral integration and two-point boundary value problems , 1991 .

[21]  S. Orszag.  Accurate solution of the Orr–Sommerfeld stability equation , 1971, Journal of Fluid Mechanics.

[22]  Fayez A. Alhargan,et al.  A Complete Method for the Computations of Mathieu Characteristic Numbers of Integer Orders , 1996, SIAM Rev..

[23]  John P. Boyd,et al.  Asymptotic coefficients of hermite function series , 1984 .

[24]  Weizhang Huang,et al.  A new pseudospectral method with upwind features , 1993 .

[25]  Peter Henrici,et al.  Essentials of numerical analysis, with pocket calculator demonstrations , 1982 .

[26]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[27]  P. G. Drazin,et al.  Solitons: Motion picture index , 1989 .

[28]  G. Strang.  A proposal for toeplitz matrix calculations , 1986 .

[29]  Wesley H. Huang,et al.  The pseudospectral method for solving di8erential eigenvalue problems , 1994 .

[30]  D. Funaro.  Polynomial Approximation of Differential Equations , 1992 .

[31]  Alex Solomonoff,et al.  Accuracy and Speed in Computing the Chebyshev Collocation Derivative , 1995, SIAM J. Sci. Comput..

[32]  Fabian Waleffe,et al.  Hydrodynamic Stability and Turbulence: Beyond Transients to a Self‐Sustaining Process , 1995 .

[33]  Richard M. Everson,et al.  On the errors incurred calculating derivatives using Chebyshev polynomials , 1992 .

[34]  Bruno Welfert.  Generation of Pseudospectral Differentiation Matrices I , 1997 .

[35]  Jean-Paul Berrut,et al.  The errors in calculating the pseudospectral differentiation matrices for C̆ebys̆ev-Gauss-Lobatto points , 1999 .

[36]  Weizhang Huang,et al.  The pseudospectral method for third-order differential equations , 1992 .

[37]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[38]  J. Weideman,et al.  Spectral Methods Based on Nonclassical Orthogonal Polynomials , 1999 .

[39]  E. Tadmor.  The exponential accuracy of Fourier and Chebyshev differencing methods , 1986 .

[40]  J. F. Botha,et al.  Error Estimates for the Solution of the Radial Schrödinger Equation by the Rayleigh—Ritz Finite Element Method , 1981 .

[41]  Tao Tang,et al.  The Hermite Spectral Method for Gaussian-Type Functions , 1993, SIAM J. Sci. Comput..