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Tau and Galerkin Methods for Fourth Order GEPs

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Spectral Methods for Non-Standard Eigenvalue Problems

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Abstract

Tau method is detailed mainly for fourth order eigenproblems. For such problems tau differentiation matrices up to fourth order are provided. As some of these problems are self-adjoint a weak (variational) along with a minimization formulation are suggested. The Galerkin method is analyzed with respect to the possibility to choice test and trial functions in order to improve the properties of the differentiation (discretization) matrices, i.e., conditioning, sparsity and symmetry. The non-normality of the differentiation (discretization) matrices is quantified using a scalar measure, i.e., the Henrici’s number and the pseudospectrum. The chapter also contains useful hints about the efficient implementation of both methods. A particular attention is paid to the capabilities of tau method to handle GEPs supplied with parameter dependent boundary conditions. The linear stability of some elastic systems as well as the linear hydrodynamic stability of some parallel shear flows (the so called Marangoni-Plateau-Gibbs effect) are analyzed in this context.

In solving a linear eigenvalue problem by a spectral method using \(N+1\) terms in spectral series, the lowest \(N/2\) eigenvalues are usually accurate to within a few percent while the larger \(N/2\) numerical eigenvalues differ from those of differential equation by such large amounts as to be useless.

Warning 1: the only reliable test is to repeat the calculations with different \(N\) and compare the results.

Warning 2: the number of good eigenvalues may be smaller than \(N/2\) if the modes have boundary layers, critical levels, or other areas of very rapid change, or when the interval is unbounded.

Boyd’s EIGENVALUE RULE-OF-THUMB [4]

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References

  1. Bjoerstad, P.E., Tjoestheim, B.P.: Efficient algorithms for solving a fourth-equation with the spectral-Galerkin method. SIAM J. Sci. Stat. Comput. 18, 621–632 (1997)

    Article  MATH  Google Scholar 

  2. Blyth, M.G., Pozrikidis, C.: Effect of surfactant on the stability of film flow down an inclined plane. J. Fluid Mech. 521, 241–250 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boyce, W.E., Di Prima, R.C.: Elementary Differential Equations and Boundary Value Problems, 9th edn. Wiley, India (2009)

    Google Scholar 

  4. Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. DOVER Publications, Inc., New York (2000)

    Google Scholar 

  5. Cabos, Ch.: A preconditioning of the tau operator for ordinary differential equations. ZAMM 74, 521–532 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Camporeale, C., Ridolfi, L.: Ice ripple formation at large Reynolds numbers. J. Fluid Mech. 694, 225–251 (2012)

    Article  MATH  Google Scholar 

  7. Camporeale, C., Canuto, C., Ridolfi, L.: A spectral approach for the stability analysis of turbulent open channel flows over granular beds. Theor. Comput. Fluid Dyn. 26, 51–80 (2012)

    Article  Google Scholar 

  8. Camporeale, C., Mantelli, E., Manes, C.: Interplay among unstable modes in films over permeable wals. J. Fluid Mech. 719, 527–550 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Canuto, C.: Spectral Methods and a Maximum Principle. Math. Comput. 51, 615–629 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chaitin-Chatelin, F., Frayssé, V.: Lectures on Finite Precision Computation. SIAM, Philadelphia (1996)

    Book  Google Scholar 

  11. Chanane, B.: Computation of the eigenvalues of Sturm-Liouville problems with parameter ependent boundary conditions using the regularized sampling method. Math. Comput. 74, 1793–1801 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Doha, E.H.: On the coefficients of differential expansions and derivatives of Jacobi polynomials. J. Phys. A: Math. Gen. 35, 3467–3478 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Doha, E.H., Bhrawy, A.H.: Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials. Appl. Numer. Math. 58, 1224–1244 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dongarra, J.J., Straughan, B., Walker, D.W.: Chebyshev tau- QZ algorithm for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22, 399–434 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. van Dorsslaer, J.L.M.: Pseudospectra for matrix pencils and stability of equilibria. BIT Numer. Math. 37, 833–845 (1997)

    Article  Google Scholar 

  16. van Dorsslaer, J.L.M.: Several concepts to investigate strongly nonnormal eigenvalue problems. SIAM J. Sci. Comput. 24, 1031–1053 (2003)

    Article  Google Scholar 

  17. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  18. Drazin, P.G., Beaumont, D.N., Coaker, S.A.: On Rossby waves modified by basic shear, and barotropic instability. J. Fluid Mech. 124, 439–456 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  19. Eberlein, P.J.: On measures of non-normality for matrices. Amer. Math. Monthly 72, 995–996 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  20. El-Daou, M.K., Ortiz, E.L., Samara, H.: A unified approach to the tau method and Chebyshev series expansion techniques. Comput. Math. Appl. 25, 73–82 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Elsner, L., Paardekooper, M.H.C.: On measure of nonnormality of matrices. Linear Algebra Appl. 92, 107–124 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fox, L., Parker, I.B.: Chebyshev Polynomials in Numerical Analysis. Oxford Mathematical Handbooks. Oxford University Press, Oxford (1968)

    Google Scholar 

  23. Funaro, D.: Polynomial Approximation of Differential Equations. Springer, Berlin Heidelberg (1992)

    MATH  Google Scholar 

  24. Gardner, D.R., Trogdon, S.A., Douglass, R.D.: A modified tau spectral method that eliminates spurious eigenvalues. J. Comput. Phys. 80, 137–167 (1989)

    Article  MATH  Google Scholar 

  25. Gheorghiu, C.I.: Spectral Methods for Differential Problems. Casa Cartii de Stiinta Publishing House, Cluj-Napoca (2007)

    MATH  Google Scholar 

  26. 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, 117–124 (1995)

    MATH  MathSciNet  Google Scholar 

  27. Gheorghiu, C.I., Pop, S.I.: A Modified Chebyshev-tau method for a hydrodynamic stability problem. In: Proceedings of ICAOR, Cluj-Napoca, vol. II, pp. 119–126 (1997)

    Google Scholar 

  28. Golub, G.H., van der Vorst, H.A.: Eigenvalue computation in the 20th century. J. Comput. Appl. Math. 123, 35–65 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications, p. 19103. SIAM, Philadelphia, Pennsilvania (1977)

    Google Scholar 

  30. Gottlieb, D., Hussaini, M.Y., Orszag, S.A.: Theory and applications of spectral methods. In: Voigt, R.G., Gottlieb, D., Hussaini, M.Y. (eds.) Spectral Methods for Partial Differential Equations, pp. 1–54. SIAM-CBMS (1984).

    Google Scholar 

  31. Greenberg, L., Marletta, M.: Numerical methods for higher order Sturm-Liouville problems. J. Comput. Appl. Math. 125, 367–383 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Greenberg, L., Marletta, M.: Numerical solution of non-self-adjoint Sturm-Liouville problems and related systems. SIAM J. Numer. Anal. 38, 1800–1845 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Hagan, J., Priede, J.: Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrdynamic stability problems by Chebyshev collocation method. arXiv:1207.0388v2[physics.com-php]. Accessed 14 Dec 2012

  34. Heinrichs, W.: A stabilized treatment of the biharmonic operator with spectral methods. SIAM J. Sci. Stat. Comput. 12, 1162–1172 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  35. Henrici, P.: Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices. Numer. Math. 4, 24–40 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  36. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York, London (1962)

    MATH  Google Scholar 

  37. Kirkner, N.P.: Computational aspects of the spectral Galerkin FEM for the Orr-Sommerfeld equation. Int. J. Numer. Meth. Fluids 32, 119–137 (2000)

    Google Scholar 

  38. Lanczos, C.: Applied Analysis. Prentice Hall Inc., Englewood Cliffs (1956)

    Google Scholar 

  39. Landau, L., Lifchitz, E.: Théorie de L’Élasticité. Édition Mir, Moscou (1967)

    MATH  Google Scholar 

  40. Lee, S.L.: A practical upper bound for departure from normality. SIAM J. Matrix Anal. Appl. 16, 462–468 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  41. Lindsay, K.A., Odgen, R.R.: A practical implementation of spectral methods resistant to the generation of spurious eigenvalues. Intl. J. Numer. Fluids 15, 1277–1294 (1992)

    Article  MATH  Google Scholar 

  42. Marletta, M., Shkalikov, A., Tretter, Ch.: Pencils of differential operators containing the eigenvalue parameter in the boundary conditions. Proc. R. Soc. Edinb. 133A, 893–917 (2003) NULL

    Google Scholar 

  43. McFaden, G.B., Murray, B.T., Boisvert, R.F.: Elimination of spurious eigenvalues in the Chebyshev tau spectral methods. J. Comput. Phys. 91, 228–239 (1990)

    Article  MathSciNet  Google Scholar 

  44. Melenk, J.M., Kirchner, N.P., Schwab, C.: Spectral Galerkin discretization for hydrodynamic stability problems. Computing 65, 97–118 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  45. Orszag, S.: Accurate solutions of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, 689–703 (1971)

    Article  MATH  Google Scholar 

  46. Ortiz, E.L.: The tau method. SIAM J. Numer. Anal. 6, 480–492 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  47. Ortiz, E.L., Samara, H.: An operational approach to the tau method for the numerical solution of non-linear differential equations. Computing 27, 15–25 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  48. Pop, I.S.: A stabilized approach for the Chebyshev-tau method. Stud. Univ. Babes-Bolyai, Math. 42, 67–79 (1997)

    Google Scholar 

  49. Pop, I.S.: A stabilized Chebyshev-Galerkin approach for the biharmonic operator. Bul. Stint. Univ. Baia-Mare Ser. B 14, 335–344 (2000)

    Google Scholar 

  50. Pop, I.S., Gheorghiu, C.I.: A Chebyshev-Galerkin method for fourth order problems. In: Proceedings of ICAOR, Cluj-Napoca, vol. II, pp. 217–220 (1997)

    Google Scholar 

  51. Roos, H.G., Pfeiffer, E.: A convergence result for the tau method. Computing 42, 81–84 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  52. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, New York (2001)

    Book  MATH  Google Scholar 

  53. Shen, J.: Efficient spectral-Galerkin method II. Direct solvers of second and fourth order equations by using Chebyshev polynomials. SIAM J. Sci. Comput. 16, 74–87 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  54. Shkalikov, A.A.: Spectral portrait of the Orr-Sommerfeld operator with large Reynolds numbers. arXiv:math-ph/0304030v1 (2003). Accessed 25 Aug 2010

    Google Scholar 

  55. Shkalikov, A.: Spectral portrait and the resolvent growth of a model problem associated with the Orr-Sommerfeld equation. arXiv:math.FA/0306342v1 (2003). Accessed 25 Aug 2010

    Google Scholar 

  56. Smith, M.K.: The mechanism for long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485 (1990)

    Article  MATH  Google Scholar 

  57. Trefethen, L.N.: Pseudospectra of linear operators. SIAM Rev. 39, 383–406 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  58. Trefethen, L.N.: Computation of pseudospectra. Acta Numerica 9, 247–295 (1999)

    Article  MathSciNet  Google Scholar 

  59. Trefethen, L.N., Trummer, M.R.: An instability phenomenon in spectral methods. SIAM J. Numer. Anal. 24, 1008–1023 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  60. Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. The Behavior of Nonnormal Matrices. Princeton University Press, Princeton and Oxford (2005)

    MATH  Google Scholar 

  61. Tretter, Ch.: A linearization for a class of \( \lambda \)-nonlinear boundary eigenvalue problems. J. Math. Anal. Appl. 247, 331–355 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  62. Tretter, Ch.: Boundary eigenvalue problems for differential equations \(N\eta =\lambda P\eta \) with \(\lambda \)-polynomial boundary conditions. Integr. J. Diff. Equat. 170, 408–471 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  63. Trif, D.: Operatorial tau method for higher order differential problems. Br. J. Math. Comput. Sci. 3, 772–793 (2013)

    Google Scholar 

  64. Zebib, A.: A Chebyshev method for the solution of boundary value problems. J. Comput. Phys. 53, 443–455 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  65. Zebib, A.: Removal of spurious modes encountered in solving stability problems by spectral methods. J. Comput. Phys. 70, 521–525 (1987)

    Article  MATH  Google Scholar 

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  1. T. Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

    Călin-Ioan Gheorghiu

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Gheorghiu, CI. (2014). Tau and Galerkin Methods for Fourth Order GEPs. In: Spectral Methods for Non-Standard Eigenvalue Problems. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06230-3_2

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