# Spectral Methods for Differential Problems

## Book summary

Summary of the book…

## Book cover

I First Part

1 Chebyshev polynomials
1.1 General properties
1.2 Fourier and Chebyshev Series
1.2.1 The trigonometric Fourier series
1.2.2 The Chebyshev series
1.2.3 Discrete least square approximation
1.2.4 Chebyshev discrete least square approximation
1.2.5 Orthogonal polynomials least square approximation
1.2.6 Orthogonal polynomials and Gauss-type quadrature formulas
1.3 Chebyshev projection
1.4 Chebyshev interpolation
1.4.1 Collocation derivative operator
1.5 Problems

2 Spectral methods for ODEs
2.1 The idea behind the spectral methods
2.2 General formulation for linear problems
2.3 Tau-spectral method
2.4 Collocation spectralmethods (pseudospectral)
2.4.1 A class of nonlinear boundary value problems
2.5 Spectral-Galerkin methods
2.6 Problems

3 Spectral methods for PDEs
3.1 Parabolic problems
3.2 Conservative PDEs
3.3 Hyperbolic problems
3.4 Problems

4 Efficient implementation
4.1 Second order Dirichlet problems for ODEs
4.2 Third and fourth order Dirichlet problems for ODEs
4.3 Problems

5 Eigenvalue problems
5.1 Standard eigenvalue problems
5.2 Theoretical analysis of a model problem
5.3 Non-standard eigenvalue problems
5.4 Problems

II Second Part

6 Non-normality of spectral approximation
6.1 A scalarmeasure of non-normality
6.2 A CG method with different trial and test basis functions
6.3 Numerical experiments
6.3.1 Second order problems
6.3.2 Fourth order problems
6.3.3 Complex Schrodinger operators

7 Concluding remarks

8 Appendix
8.1 Lagrangian and Hermite interpolation
8.2 Sobolev spaces
8.2.1 The Spaces Cm¡Ω¢, m ≥ 0
8.2.2 The Lebesgue Integral and Spaces Lp (a, b) , 1 ≤ p ≤ ∞
8.2.3 Infinite Differentiable Functions and Distributions
8.2.4 Sobolev Spaces and Sobolev Norms
8.2.5 TheWeighted Spaces
8.3 MATLAB codes

keyword1,

## References

see the expanding block below

##### Cite this book as:

C.I. Gheorghiu, Spectral Methods for Differential Problems, Casa Cărtii de Stiintă, Cluj-Napoca, Romania, 2007.

?

?

?

##### Topics

The book on google scholar.

[1] Ablowitz, M.J., Herbst, B.M., Schober, C., On the Numerical Solution of the Sine-Gordon Equation, I Integrable Discretizations and Homoclinic Manifolds, J. Comput. Phys., 126(1996), 299-314
[2] Ache, G.A., Cores, D., Note on the Two-Point Boundary Value Numerical Solution of the Orr-Sommerfeld Stability Equation, J. Comput. Phys., 116(1995), 180-183
[3] Adams, R. A., Sobolev Spaces, Academic Press, New York-San FranciscoLondon, 1975
[4] Andrew, A. L., Centrosymmetric Matrices, SIAM Rev., 40(1998), 697-698
[5] Andrews, L. C., Special Functions of Mathematics for Engineers, International Edition 1992, McGraw-Hill, Inc. 1992
[6] Antohe, V., Gladwell, I., Performance of two methods for solving separable Hamiltonian systems, J. Comput. Appl. Math., 125(2000), 83-92
[7] Ascher, U. M., Russel, R. D., Reformulation of Boundary Value Problems into ”Standard” Form, SIAM Rev. 23(1981), 238-254
[8] Ascher, U. M., Mattheij, R, M. M., Russel, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, 1988
[9] Ascher, U. M., McLachlan, R. I., Multisymplectic box schemes and the Korteweg-de Vries equation, Preprint submitted to Elsevier Sciences, 31 July, 2003
[10] Atkinson, K.E., Elementary Numerical Analysis, John Wiley &Sons, 1985
[11] Babuska, I., Aziz, K., Survey Lectures on the Mathematical Foundation of the Finite Element Method, in The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, Ed. A. K. Aziz, Academic Press, London/New York 1972, pp. 3-359
[12] Basdevant, C., Deville, M., Haldenwang, P., Lacroix, J. M., Ouazzani, J., Peyret, R., Orlandi, P., Patera, A.T., Spectral and Finite Solutions of the Burgers Equation, Computers and Fluids, 14(1986) 23-41
[13] Berland, H., Islas, A. L., Schoder, C., Conservation of phase properties using exponential integrators on the cubic Schr¨odinger equation, Preprint Norvegian University of Sciences and Technology, Trondheim, Norway, 2006
[14] Bernardi, C., Maday, Y., Approximations Spectrales de Problems aux Limites Elliptiques, Springer Verlag, Paris, 1992
[15] Birkhoff, G., Rota, G.-C., Ordinary Differential Equations, John Wiley & Sons, Second Edition, 1969
[16] Bjoerstad, P.E., Tjoestheim, B.P., Efficient algorithms for solving a fourth-equation with the spectral-Galerkin method, SIAM J. Sci. Stat. Comput. 18(1997), 621-632
[17] Boyd, J. P., Numerical Computations of a Nearly Singular Nonlinear Equation: Weakly Nonlocal Bound States of Solitons for the Fifth-Order Korteweg-de Vries Equation, J. Comput. Phys. 124(1996), 55-70
[18] Boyd, J. P., Traps and Snares in Eigenvalue Calculations with Application to Pseudospectral Computations of Ocean Tides in a Basin Bounded by Meridians, J. Comput. Phys., 126(1996), 11-20
[19] Boyd, J. P., Chebyshev and Fourier Spectral Methods, Second Edition, DOVER Publications, Inc., 2000
[20] Boyd, J. P., Trouble with Gegenbauer reconstruction for defeating Gibbs’ phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations, J. Comput. Phys., 204(2005), 253-264
[21] Boyd, J. P., A Chebyshev/rational Chebyshev spectral method for the Helmholtz equation in a sector on the surface of a sphere: defeating corner singularities, J. Comput. Phys., 206(2005), 302-310
[22] Breuer, K.S., Everson, R.M., On the Errors Incurred Calculating Derivatives Using Chebyshev Polynomials, J. Comput. Phys., 99(1992), 56-67
[23] Brusch, L., Torcini, A., van Hecke, M., Zimmermann, M. G., B¨ar, M., Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation, Physica D, 160(2001) 127-148
[24] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations-Runge-Kutta and General Linear Methods, John Wiley &Sons, 1987
[25] Butcher, J. C., Numerical methods for ordinary differential equations in the 20th century, J. Comput. Appl. Math., 125(2000), 1-29
[26] Butzer, P., Jongmans, F., P. L. Chebyshev (1821-1894) A Guide to his Life and Work, J. Approx. Theory, 96(1999) 111-138
[27] Cabos, Ch., A preconditioning of the tau operator for ordinary differential equations, ZAMM 74(1994) 521-532
[28] Canuto, C., Boundary Conditions in Chebyshev and Legendre Methods, SIAM J. Numer. Anal., 23(1986) 815-831
[29] Canuto, C., Spectral Methods and a Maximum Principle, Math. Comput., 51(1988), 615-629
[30] Canuto, C., Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38(1982), 67-86
[31] Canuto, C., Quarteroni, A., Variational Methods in the Theoretical Analysis of Spectral Approximations, in Spectral Methods for Partial Differential Equations, Ed. by R. G. Voigt, D. Gottlieb, M. Y. Hussaini, SIAM-CBMS, P. 55-78
[32] Canuto, C., Quarteroni, A., Spectral and pseudo-spectral methods for parabolic problems with non-periodic boundary conditions, Calcolo 18(1981), 197-218
[33] Canuto, C, Hussaini, M.Y., Quarteroni, A., Zang, T.A., Spectral Methods in Fluid Dynamics, Springer Verlag, 1987
[34] Carpenter, M. H., Gottlieb, D., Spectral Methods on Arbitrary Grids, J. Comput. Phys., 129(1996), 74-86
[35] Coron, J-M, Crepeau, E, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc., 6(2004) 367-398
[36] Chaitin-Chatelin, F. and V. Fraysse, Lectures on Finite Precision Computation, SIAM Philadelphia, 1996
[37] Chan, T. F., Kerkhoven, T., Fourier Methods with Extended Stability Intervals for the Korteweg-De Vries Equation, SIAM J. Numer. Anal., 22(1985), 441-454
[38] Ciarlet, P.G., The Finite Element Method for Elliptic Equations, NorthHolland Publishing Company, 1978
[39] Ciarlet, P.G., Introduction to Numerical Linear Algebra and Optimization, Cambridge Texts in Applied Mathematics, CUP, 1989
[40] Cooley, J. W., Tukey, J. W., An Algorithm for the Machine Calculation of Complex Fourier Series, Math. Comput., 19(1965) 297-301
[41] Cooley, J. W., Lewis, P. A. W., Welch, P. D., The Fast Fourier Transform Algorithm: Programming Considerations in the Calculation of Sine, Cosine, and Laplace Transforms, J. Sound. Vib., 12(1970) 315-337
[42] Davis, P.J., Interpolation and Approximation, Blaisdell Pub. Co., NewYork, 1963
[43] Davis, P.J., Rabinowitz, P., Methods of Numerical Integration, New York: Academic Press, 1975
[44] Deeba, E., Khuri, S. A., A decomposition Method for Solving the Nonlinear Klein-Gordon Equation, J. Comput. Phys., 124(1996), 442-448
[45] Deeba, E., Khuri, S. A., Xie, S., An Algorithm for Solving Boundary Value Problems, J. Comput. Phys., 159(2000), 125-138
[46] Dendy, J. E. Jr., Galerkin’s Method for some Highly Nonlinear Problems, SIAM J Numer. Anal., 14(1997), 327-347
[47] Denis, S.C.R., Quartapelle, L., Spectral Algorithms for Vector Elliptic Equations in a Spherical Gap, J. Comput. Phys., 61(1985), 218-241
[48] Deuflhard, P., Hohmann, A., Numerical Analysis in Modern Scientific Computing; An Introduction, Springer Verlag, 2003
[49] Doha, E.H., Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. Algor., 42(2006), 137-164
[50] Doha, E.H., Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math., 58(2008), 1224-1244
[51] Doha, E.H., Abd-Elhameed, W. M., Bhrawy, A.H., Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2n th-order linear differential equations, Appl. Math. Modell., 33(2009), 1982-1996
[52] Doha, E.H., Abd-Elhameed, W. M., Efficient spectral ultraspherical-dualPetrov-Galerkin algorithms for the direct solution of (2n+1) th-order linear differential equations, Math. Comput. Simul., 79(2009), 3221-3242
[53] Doha, E.H., Bhrawy, A.H., Abd-Elhameed, W. M., Jacobi spectral Galerkin method for elliptic Neumann problems, Numer. Algor., 50(2009), 67-91
[54] Don, W. S., Gottlieb, D., The Chebyshev-Legendre Method: Implementing Legendre Methods on Chebyshev Points, SIAM J. Numer. Anal., 31(1994), 1519-1534
[55] Dongarra, J.J., Straughan, B., Walker, D.W., Chebyshev tau- QZ algorithm for calculating spectra of hydrodynamic stability problems, Appl. Numer. Math. 22(1996), 399-434
[56] Drazin, P.G., Nonlinear Systems, Cambridge University Press, 1992
[57] Drazin, P.G., Beaumont, D.N., Coaker, S.A., On Rossby waves modified by basic shear, and barotropic instability, J. Fluid Mech. 124(1982), 439-456
[58] Driscoll, T. A., Fornberg, B., A Block Pseudospectral Method for Maxwell’s Equations, J. Comput. Phys., 140(1998), 47-65
[59] Eberlein, P. J., On measures of non-normality for matrices, Amer. Math. Monthly 72(1965), 995-996
[60] Elbarbary, E. M. E., Ei-Sayed, S. M., Higher order pseudospectral differentiation matrices, Appl. Numer. Math., 55(2005), 425-438
[61] El-Daou, M.K., Ortiz, E.L., Samara, H., A Unified Approach to the Tau Method and Chebyshev Series Expansion Techniques, Computers Math. Applic. 25(1993), 73-82
[62] El-gamel, M., A comparison between the Sinc-Galerkin and the modified decomposition methods for solving two-point boundary-value problems, J. Comput. Phys., 223(2007), 369-383
[63] L. Elsner, M. H. C. Paardekooper, On Measure of Nonnormality of Matrices, Linear Algebra Appl. 92:107-124 (1897)
[64] Engquist, B., Osher, S., One-Sided Difference Approximations for Nonlinear Conservation Laws, Math. Comp. 36(1981) 321-351
[65] Fatone, L., Funaro, D., Yoon, G. J., A convergence analysis for the superconsistent Chebyshev method, Appl. Numer. Math., 58(2008), 88-100
[66] Fishelov, D., The Spectrum and the Stability of the Chebyshev Collocation Operator for Transonic Flow, Math. Comput. 51(1988), 559-579
[67] Fornberg, B., On the Instability of Leap-Frog and Crank-Nicolson Approximations of a Nonlinear Partial Differential Equation, Math. Comput. 27(1973) 45-57
[68] Fornberg, B., Generation of Finite Difference Formulas on Arbitrary Spaced Grids, Math. Comput., 51(1988), 699-706
[69] Fornberg, B., Sloan, D., M., A review of pseudospectral methods for solving partial differential equations, Acta Numerica, 1994, 203-267
[70] Fornberg, B., Driscoll, T.A., A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion, J. Comput. Phys. 155(1999), 456-467
[71] Fox, L., Parker, I.B., Chebyshev Polynomials in Numerical Analysis, Oxford Mathematical Handbooks, O U P, 1968
[72] Funaro, D., A Preconditioning Matrix for the Chebyshev Differencing Operator, SIAM J. Numer. Anal., 24(1987) 1024-1031
[73] Funaro, D., FORTRAN routines for spectral methods, http://cdm.unimo.it/home/matematica/funaro.daniele/fman.pdf
[74] Funaro, D., A Variational Formulation for the Chebyshev Pseudospectral Approximation of Neumann Problems, SIAM J. Numer. Anal. 27(1990), 695-703
[75] Funaro, D., A New Scheme for the Approximation of Advection-Diffusion Equations by Collocation, SIAM J. Numer. Anal., 30(1993), 1664-1676
[76] Funaro, D., Heinrichs,W., Some results about the pseudospectral approximation of one-dimensional fourth-order problems, Numer. Math. 58(1990), 399-418
[77] Funaro, D., Kavian, O., Approximation of Some Diffusion Evolution Equations in Unbounded Domains by Hermite Functions, Math. Comput. 57(1991), 597-619
[78] Garcia-Archilla, B., A Spectral Method for the Equal Width Equation, J. Comput. Phys., 125(1996), 395-402
[79] Gardner, C. S. Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vrie Equation as a Hamiltonian System, J. Math. Phys., 12(1971), 1548-1561
[80] Gardner, D. R., Trogdon, S. A., Douglass, R. D., A Modified Tau Spectral Method That Eliminates Spurious Eigenvalues, J. Comput. Phys., 80(1989)137-167
[81] Gheorghiu, C.I., Pop, S.I.,On the Chebyshev-tau approximation for some singularly perturbed two-point boundary value problems, Rev. Roum. Anal. Numer. Theor. Approx., 24(1995), 117-124, Zbl M 960.44077
[82] Gheorghiu, C.I., Pop, S.I., A Modified Chebyshev-tau Method for a Hydrodynamic Stability Problem, Proceedings of I C A O R, vol. II, pp.119-126, Cluj-Napoca, 1997 MR 98g:41002
[83] Gheorghiu, C.I., A Constructive Introduction to Finite Elements Method, Qvo Vadis, Cluj-Napoca, 1997
[84] Gheorghiu, C. I., On the Scalar Measure of Non-Normality of Matrices; Dimension vs. Structure, General Mathematics, U. L. B. Sibiu, 11:21-32 (2003)
[85] Gheorghiu, C. I., On the spectral Characterization of some ChebyshevType Methods; Dimension vs. Structure, Studia Univ. ”Babes-Bolyai”, Mathematica, L(2005) 61-66
[86] Gheorghiu, C. I., Trif, D., The numerical approximation to positive solution of some reaction-diffusion problems, PU. M. A., 11(2000) 243-253
[87] Golub, G. H., Wilkinson, J. H., Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form, SIAM Review, 18(1976) 578-619
[88] Golub, G. H., Ortega, J. M., Scientific Computing and Differential Equations-An Introduction to Numerical Methods, Academic Press, 1992
[89] Golub, G. H., van der Vorst, H. A., Eigenvalue computation in the 20th century, J. Comput. Appl. Math., 123(2000), 35-65
[90] Gottlieb, D., Orszag, S.A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, 1977
[91] Gottlieb, D., Turkel, E., On Time Discretization for Spectral Methods, Studies in Appl. Math., 63(1980), 67-86
[92] Gottlieb, D., The Stability of Pseudospectral-Chebyshev Methods, Math. Comp., 36(1981), 107-118
[93] Gottlieb, D., Hussaini, M.Y., Orszag, S.A., Theory and Applications of Spectral Methods, in Spectral Methods for Partial Differential Equations, Ed. by R.G. Voigt, D. Gottlieb, M.Y. Hussaini, SIAM-CBMS, P. 1-54, 1984
[94] Gottlieb, D., Orszag, S.A., Turkel, E., Stability of Pseudospectral and Finite-Difference Methods for Variable Coefficient Problem, Math. Comp. 37(1981), 293-305
[95] Gottlieb, D., Hesthaven, J. S., Spectral methods for hyperbolic problems, J. Comput. Appl. Math., 128(2001), 83-131
[96] Greenberg, L., Marleta, M., Numerical methods for higher order SturmLiouville problems, J. Comput. Appl. Math., 125(2000) 367-383
[97] Greenberg, L., Marleta, M., Numerical Solution of Non-Self-Adjoint Sturm-Liouville Problems and Related Systems, SIAM J. Numer. Anal., 38(2001), 1800-1845
[98] Greengard, L., Rokhlin, V., On the Numerical Solution of Two-Point Boundary Value Problems, Comm. Pure Appl. Math. XLIV(1991), 419-452
[99] Guo, B.-y, Wang, Z.-q, Wan, Z.-s, Chu, D., Second order Jacobi approximation with applications to fourth-order differential equations, Appl. Numer. Math., 55(2005), 480-502
[100] Hamming, R. W., Introduction to Applied Numerical Analysis, International Student Edition, McGraw-Hill, Inc. 1971
[101] Hairer, E., Hairer, M., GniCodes-Matlab Programs for Geometric Numerical Integration, http://www.unige.ch/math/folks/hairer
[102] Heinrichs, W., Spectral Methods with Sparse Matrices, Numer. Math., 56(1989), 25-41
[103] Heinrichs, W., Improved Condition Number for Spectral Methods, 53(1998), Math. Comp., 103-119
[104] Heinrichs, W., Stabilization Techniques for Spectral Methods, J. Sci. Comput., 6(1991), 1-19
[105] Heinrichs, W., A Stabilized Treatment of the Biharmonic Operator with Spectral Methods, SIAM J. Sci. Stat. Comput., 12(1991), 1162-1172
[106] Heinrichs, W., Strong Convergence Estimates for Pseudospectral Methods, Appl. Math., (1992), 401-417
[107] Heinrichs, W., Spectral Approximation of Third-Order Problems, J. Scientific Computing, 14(1999), 275-289
[108] Henrici,P., Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math. 4, 24-40(1962)
[109] Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London, 1962
[110] Henrici, P., Essentials of Numerical Analysis with Pocket Calculator Demonstrations, John Wiley & Sons, Inc., New York, 1982
[111] Hiegemann, M., Strauss, K., On a Chebyshev matrix operator method for ordinary differential equations with non-constant coefficients, Acta Mech., 105(1994), 227-232
[112] Hiegemann, M., Chebyshev matrix operator method for the solution of integrated forms of linear ordinary differential equations, Acta Mech. 122(1997), 231-242
[113] Higham, D. J., Owren, B., Nonnormality Effects in a Discretised Nonlinear Reaction-Convection-Diffusion Equation, J. Comput. Phys., 124(1996), 309-323
[114] Hill, A. A., Straughan, B., Linear and non-linear stability thresholds for thermal convection in a box, Math. Meth. Appl. Sci. 29(2006), 2123-2132
[115] Holden, H., Karlsen, K. H., Risebro, N. H., Operator Splitting Methods for Generalized Korteweg-De Vries Equations, J. Comput. Phys. 153(1999), 203-222
[116] Huang, W., Sloan, D.M., The pseudospectral method for third-order differential equations, SIAM J. Numer. Anal. 29(1992), 1626-1647
[117] Huang, W., Sloan, D.M., The pseudospectral method for solving differential eigenvalue problems, J. Comput. Phys. 111(1994), 399-409
[118] Hunt, B. R., Lipsman, R. L., Rosenberg, J. M., Coombes, K. R., Osborn, J. E., Stuck, G. J., A guide to MATLAB for Beginners and Experienced Users, Cambridge University Press, 2001
[119] Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996
[120] Ismail, M. S., Numerical solution of coupled nonlinear Schr¨odinger equation by Galerkin method, Math. Comput. Simul. 78(2008), 532-547
[121] Johnson, C., Numerical solutions of partial differential equations by the finite element method, Cambridge University Press, 1987
[122] Jung, J-H, Shizgal, B. D., On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon, J. Comput. Phys., 224(2007), 477-488
[123] Kosugi, S., Morita, Y., Yotsutani, S., Complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions, Communications on Pure and Applied Analysis, 4(2005) 665-682
[124] R. Kress, H. L. de Vries, R. Wegmann, On Nonnormal Matrices, Linear Algebra Appl. 8, 109-120(1974)
[125] Kreiss, H-O., Oliger, J., Stability of the Fourier Method, SIAM J. Numerical Analysis, 16(1979), 421-433
[126] Lanczos, C., Applied Analysis, Prentice Hall Inc., Englewood Cliffs, N. J.,1956
[127] Lax, P.D., Milgram, A.N., Parabolic equations, Annals of Math. Studies, No. 33(1954) Princeton Univ. Press
[128] Lax, P.D., Integrals of Nonlinear Equations of Evolution and Solitary Waves, Communications on Pure and Applied Mathematics, XXI(1968), 467-490
[129] Lax, P.D., Almost periodic solutions of the KdV equation, SIAM Rev., 18(1976) 351-375
[130] Lee, S. L., A Practical Upper Bound for Departure from Normality, SIAM J. Matrix Anal. Appl., 16, 462-468, April 1995
[131] Lindsay, K.A., Odgen, R.R., A Practical Implementation of Spectral Methods Resistant to the Generation of Spurious Eigenvalues, Intl. J. Numer. Fluids 15(1992), 1277-1294
[132] Maday, Y., Quarteroni, A., Legendre and Chebyshev Spectral Approximations of Burgers’ Equation, Numer. Math. 37(1981) 321-332
[133] Maday, Y., Analysis of Spectral Projectors in One-Dimensional Domains, Math. Comp. 55(1990), 537-562
[134] Maday, Y., Metivet, B., Chebyshev Spectral Approximation of NavierStokes Equations in a Two Dimensional Domain, Model Math. Anal. Numer. (M2AN)21(1987), 93-123
[135] Malik, S.V., Stability of the interfacial flows, Technical Report, University of the West of England, Bristol-England, 2003
[136] Markiewicz, D., Survey on Symplectic Integrators, Preprint Univ. California at Berkeley, Spring 1999
[137] Matano, H., Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. (JMKYAZ) 18-2(1978), 221-227
[138] McFaden, G.B., Murray, B.T., Boisvert, R. F., Elimination of Spurious Eigenvalues in the Chebyshev Tau Spectral Methods, J. Comput. Phys.,91(1990) 228-239
[139] McLachlan, R., Symplectic integration of Hamiltonian wave equations, Numer. Math., 66(1994), 465-492
[140] Mead, J. L., Renaut, R. A., Optimal Runge-Kutta Methods for First Order Pseudospectral Operators, J. Comput. Phys., 152(1999), 404-419
[141] Melenk, J.M., Kirchner, N.P., Schwab, C., Spectral Galerkin Discretization for Hydrodynamic Stability Problems, Computing 65(2000), 97-118
[142] Miele, A., Aggarwal, A. K., Tietze, J. L., Solution of Two-Point Boundary-Value Problems with Jacobian Matrix Characterized by Large Positive Eigenvalues, J. Comput. Phys., 15(1974), 117-133
[143] Mitchell, A.R., Murray, B.A., Sleeman, B.D., Numerical Solution of Hamiltonian Systems in Reaction-Diffusion by Symplectic Difference Schemes, J. Comput. Phys., 95(1991), 339-358
[144] Miura, R., The Korteweg-De Vries Equation: A Survey of Results, SIAM Rev., 18(1976), 412-459
[145] Monro, D. M., Interpolation by Fast Fourier and Chebyshev Transforms, Int. J. for Num. Met. in Engineering, 14(1979) 1679-1692
[146] Moser, J., Recent developments in the theory of Hamiltonian systems, SIAM Rev., 28(1986), 459-485
[147] Murty, V. N., Best Approximation with Chebyshev Polynomials, SIAM J. Numer. Anal., 8(1971), 717-721
[148] Necas, J., Sur une methode pour resoudre les equations aux derivee partielles du type elliptiques, voisine de la variationelle, Ann. Sc. Norm. Sup. Pisa 16(1962), 305-326
[149] Nield, D. A., Odd-Even Factorization Results for Eigenvalue Problems, SIAM Rev., 36(1994), 649-651
[150] Nikolsky, S.M., A Course of Mathematical Analysis, Mir Publishers Moscow, 1981.
[151] Omelyan, I. P., Mryglod, I. M., Folk, R., Molecular dynamics simulations of spin and pure liquids with preservation of all the conservation laws, Phys. Rev. E 64, 016105(2001)
[152] Omelyan, I. P., Mryglod, I. M., Folk, R., Construction of high-order forcegradient algorithms for integration of motion in classical and quantum systems, Phys. Rev. E 66, 026701(2002)
[153] Omelyan, I. P., Mryglod, I. M., Folk, R., Optimized Verlet-like algorithms for molecular dynamics simulations, Phys. Rev. E 65, 056706(2002)
[154] O’Neil, P.V., Advanced Engineering Mathematics, International Student Edition, Chapman & Hall, 3rd. edition, 1991
[155] Orszag, S., Accurate solutions of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50(1971), 689-703
[156] Orszag, S., Comparison of Pseudospectral and Spectral Approximations, Studies in Appl. Math. 51(1971), 253-259
[157] Ortiz, E.L., The Tau method, SIAM J. Numer. Anal., 6(1969), 480-492
[158] Ortiz, E.L., Samara, H., An Operational Approach to the Tau Method for the Numerical Solution of Non-Linear Differential Equations, Computing, 27(1981), 15-25
[159] OuldKaber, S.M., A Legendre Pseudospectral Viscosity Method, J. Comput. Phys., 128(1996), 165-180
[160] Parter, S. V., Rothman, E. E., Preconditioning Spectral Collocation Approximations to Elliptic Problems, SIAM J. Numer. Anal. 32(1995) 333-385
[161] Petrila, T., Trif, D., Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics, Kluwer Academic Publishers, Boston/Dordrecht/London, 2004
[162] Pichmony Anhaouy, Fourier Spectral Methods for Solving the Korteweg-de Vries Equation, Thesis, Simon Fraser University, 2000
[163] Pop, I.S., Numerical Approximation of Differential Equations by Spectral Methods, Technical report, ”Babes-Bolyai” University, Cluj-Napoca, 1995 (in Romanian)
[164] Pop, I.S., Gheorghiu, C.I., A Chebyshev-Galerkin Method for Fourth Order Problems, Proceedings of I C A O R, vol. II, pp.217-220, 1997 MR 98g:41002
[165] Pop, I.S., A stabilized approach for the Chebyshev-tau method, Studia Univ. ”Babes-Bolyai”, Mathematica, 42(1997), 67-79
[166] Pruess, S., Fulton, C. T., Mathematical Software for Sturm-Liouville Problems, A C M Trans. Math. Softw. 19(1993), 360-376
[167] Pryce, J. D., A Test Package for Sturm-Liouville Solvers, A C M Trans. on Math. Software, 25(1999), 21-57
[168] Qiu, Y., Sloan, D. M., Numerical Solution of Fischer’s Equation Using a Moving Mesh Method, J. Comput. Phys., 146(1998), 726-746
[169] Quarteroni, A., Valli, A., Numerical Approximation of Partial Differential Equations, Springer Verlag, Berlin/Heidelberg, 1994
[170] Quarteroni, A., Saleri, F., Scientific Computing with MATLAB and Octave, Second Ed., Springer, 2006
[171] Raltson, A., Rabinowitz, Ph., A First Course in Numerical Analysis, McGraw Hill, 1978
[172] Roos, H.G., Pfeiffer, E., A Convergence Result for the Tau Method, Computing 42(1989), 81-84
[173] Shampine, L. F., Reichelt, M. W., The MATLAB O D E Suite, SIAM J. Sci. Comput., 18(1997), 1-22
[174] Shampine, L. F., Design of software for ODE, J. Comput. Appl. Math., 205(2007),
[175] Shen, J., Efficient spectral-Galerkin method I, Direct solvers of second and fourth equations using Legendre polynomials, SIAM J. Sci. Stat. Comput., 15(1994), 1489-1505
[176] Shen, J., Efficient Spectral-Galerkin Method II. Direct Solvers of Second and Fourth Order Equations by Using Chebyshev Polynomials, SIAM J. Sci. Comput., 16(1995), 74-87
[177] Shen, J., Efficient Chebyshev-Legendre Galerkin methods for elliptic problems, Proceedings of ICOSAHOM’95, Houston J. Math. (1996) 233-239
[178] Shen, J., Temam, R., Nonlinear Galerkin Method Using Chebyshev and Legendre Polynomials I. The One-Dimensional Case, SIAM J. Numer. Anal. 32(1995) 215-234
[179] Shkalikov, A.A., Spectral portrait of the Orr-Sommerfeld operator with large Reynolds numbers, arXiv:math-ph/0304030v1, 22Apr2003
[180] Shkalikov, A.A., Spectral portrait and the resolvent growth of a model problem associated with the Orr-Sommerfeld equation, arXiv:math.FA/0306342v1, 24Jun2003
[181] Simmons, G. F., Differential Equations with Applications and Historical Notes, McGraw-Hill Book Company, 1972
[182] Sloan, D.M., On the norms of inverses of pseudospectral differential matrices, SIAM J. Numer. Anal., 42(2004), 30-48
[183] Solomonoff, A., Turkel, E., Global Properties of Pseudospectral Methods, J. Comput. Phys., 81(1989) 239-276
[184] Solomonoff, A., A Fast Algorithm for Spectral Differentiation, J. Comput. Phys., 98(1992), 174-177
[185] Stenger, F., Numerical Methods Based on Whittaker Cardinal, or Sinc Functions, SIAM Rev. 23(1981), 165-224
[186] Stenger, F., Summary of Sinc numerical methods, J. Comput. Appl. Math., 121(2000), 379-420
[187] Stoer, J., Bulirsch, R., Introduction to Numerical Analysis, Springer Verlag, New York, Heidelberg, Berlin, 1980
[188] B. J. Stone, Best possible ratios of certain matrix norms, Numer. Math. 4,114-116(1962)
[189] Tadmor, E., The Exponential Accuracy of Fourier and Chebyshev Differencing Methods, SIAM J. Numer. Anal., 23(1986), 1-10
[190] Tadmor, E., Stability Analysis of Finite-Difference, Pseudospectral and Fourier-Galerkin Approximations for Time-Dependent Problems, SIAM Rev., 29(1987), 525-555
[191] Tal-Ezer, H., A Pseudospectral Legendre Method for Hyperbolic Equations with an Improved Stability Condition, J. Comput. Phys., 67(1986), 175-172
[192] Tang, T., The Hermite spectral method for Gaussian-type functions, SIAM J. Sci. Comput. 14(1993), 594-606
[193] Trefethen, L. N., Pseudospectra of linear operators, SIAM Review, 39(1997)
[194] Trefethen, L. N., Computation of Pseudospectra, Acta Numerica, 247-295(1999)
[195] Trefethen, L. N., Reichel, L., Eigenvalues and Pseudoeigenvalues of Toeplitz Matrices, Linear Algebra Appl., 162-164:153-158(1992)
[196] Trefethen, L. N., Trummer, M. R., An Instability Phenomenon in Spectral Methods, SIAM J. Numer. Anal. 24(1987), 1008-1023.
[197] Trefethen, L. N., Spectral Methods in MATLAB, SIAM, Philadelphia, PA, 2000
[198] Trefethen, L. N., Embree, M., Spectra and Pseudospectra; The Behavior of Nonnormal Matrices, Princeton University Press, Princeton and Oxford, 2005
[199] Tretter, Ch., A Linearization for a Class of λ−Nonlinear Boundary Eigenvalue Problems, J. Mathematical Analysis and Applications, 247(2000), 331-355
[200] van Saarloos, W., The Complex Ginzburg-Landau equation for beginners, in Spatio-temporal Patterns in Nonequilibrium Complex Systems, eds. P.E. Cladis and P. Palffy-Muhoray, Santa Fe Institute, Studies in the Science of Complexity, proceedings XXI, Addison-Wesley, Reading, 1994
[201] Varga, R. S., On Higher Order Stable Implicit Methods for Solving Parabolic Partial Differential Equations, J. Math. and Phys., XL (1961), 220-231
[202] Venakides, S., Focusing Nonlinear Schroedinger equation: Rigorous Semiclassical Asymptotics, http://www.iacm.forth.gr/anogia05/Docs/venakides material.pdf
[203] Weideman, J.A.C., Trefethen, L.N., The Eigenvalues of Second-Order Spectral Differentiations Matrices, SIAM J. Numer. Anal. 25(1988), 1279-1298
[204] Weideman, J.A.C., Reddy, S. C., A MATLAB Differentiation Matrix Suite, ACM Trans. on Math. Software, 26(2000), 465-519
[205] Weinan, E., Convergence of spectral methods for Burger’s equation, SIAM J. Numer. Anal., 29(1992), 1520-1541
[206] Welfert, B. D., Generation of pseudospectral differentiation matrices, SIAM J. Numer. Anal., 34(1997), 1640-1657
[207] Wright, T. G., Trefethen, L. N., Large-Scale Computation of Pseudospectra Using ARPACK and EIGS, SIAM J. Sci. Comput., 23(2001), 591-605
[208] Wright, T. G., http://www.comlab.ox.ac.uk/oucl/work/tom.wright/psgui/, University of Oxford, 2000
[209] Wu, X., Kong, W., Li, C., Sinc collocation method with boundary treatment for two-point boundary value problem, J. Comput. Appl. Math., 196(2006), 229-240
[210] Zebib, A., A Chebyshev Method for the Solution of Boundary Value Problems, J. Comput. Phys., 53(1984), 443-455
[211] Zebib, A., Removal of Spurious Modes Encountered in Solving Stability Problems by Spectral Methods, J. Comput. Phys., 70(1987), 521-525