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Solving differential eigenproblems via the spectral Tau method

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Abstract

The spectral Tau method to compute eigenpairs of ordinary differential equations is implemented as part of the Tau Toolbox—a numerical library for the solution of integro-differential problems. This mathematical software enables a symbolic syntax to be applied to objects to manipulate and solve differential problems with ease and accuracy. The library is explained in detail and its application to various problems is illustrated: numerical approximations for linear, quadratic, and nonlinear differential eigenvalue problems.

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Notes

  1. http://twr.cs.kuleuven.be/research/software/nleps/nleigs.php

References

  1. Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., Van Der Vorst, H.: Templates for the solution of algebraic eigenvalue problems: a practical guide. SIAM (2000). https://doi.org/10.1137/1.9780898719581

  2. Bailey, P.B., Everitt, W.N., Zettl, A.: Algorithm 810: the sleign2 Sturm-Liouville code. ACM Trans. Math. Softw. 27(2), 143–192 (2001). https://doi.org/10.1145/383738.383739

    Article  MATH  Google Scholar 

  3. Boyd, J.: Chebyshev and Fourier spectral methods. Dover publications Inc (2000)

  4. Bridges, T.J., Morris, P.J.: Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55(3), 437–460 (1984). https://doi.org/10.1016/0021-9991(84)90032-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Butler, K.M., Farrell, B.F.: Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A-Fluid 4(8), 1637–1650 (1992). https://doi.org/10.1063/1.858386

    Article  Google Scholar 

  6. Charalambides, M., Waleffe, F.: Gegenbauer Tau methods with and without spurious eigenvalues. SIAM J. Numer. Anal. 47(1), 48–68 (2009). https://doi.org/10.1137/070704228

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaves, T., Ortiz, E.L.: On the numerical solution of two-point boundary value problems for linear differential equations. Z Angew Math. Mech. 48(6), 415–418 (1968). https://doi.org/10.1002/zamm.19680480607

    Article  MathSciNet  MATH  Google Scholar 

  8. Dawkins, P.T., Dunbar, S.R., Douglass, R.W.: The origin and nature of spurious eigenvalues in the spectral Tau method. J. Comput. Phys. 147 (2), 441–462 (1998). https://doi.org/10.1006/jcph.1998.6095

    Article  MathSciNet  MATH  Google Scholar 

  9. Dongarra, J., Straughan, B., Walker, D.: Chebyshev Tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Math. 22(4), 399–434 (1996). https://doi.org/10.1016/S0168-9274(96)00049-9

    Article  MathSciNet  MATH  Google Scholar 

  10. Driscoll, T.A., Hale, N.: Rectangular spectral collocation. IMA J. Numer. Anal. 36(1), 108–132 (2016). https://doi.org/10.1093/imanum/dru062

    MathSciNet  MATH  Google Scholar 

  11. Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide (2014)

  12. Gardner, D.R., Trogdon, S.A., Douglass, R.W.: A modified Tau spectral method that eliminates spurious eigenvalues. J. Comput. Phys. 80(1), 137–167 (1989). https://doi.org/10.1016/0021-9991(89)90093-4

    Article  MATH  Google Scholar 

  13. Gheorghiu, C.I.: Accurate spectral collocation computation of high order eigenvalues for singular Schrödinger equations. Computation 9(2), 2 (2021). https://doi.org/10.3390/computation9010002

    Google Scholar 

  14. Gheorghiu, C.I., Pop, I.S.: A modified chebyshev-tau method for a hydrodynamic stability problem. In: Stancu, D.D. (ed.) Approximation and optimization. Transilvania Press, Cluj-Napoca, pp. 119–126 (1996)

  15. Gheorghiu, C.I., Rommes, J.: Application of the Jacobi-Davidson method to accurate analysis of singular linear hydrodynamic stability problems. Int. J. Numer. Methods Fluids 71(3), 358–369 (2012). https://doi.org/10.1002/fld.3669

    Article  MathSciNet  MATH  Google Scholar 

  16. Greenberg, L., Marletta, M.: Algorithm 775: the code SLEUTH for solving fourth-order Sturm-Liouville problems. ACM T. Math. Softw. 23(4), 453–493 (1997). https://doi.org/10.1145/279232.279231

    Article  MathSciNet  MATH  Google Scholar 

  17. Güttel, S., Tisseur, F.: The nonlinear eigenvalue problem. Acta Numerica 26, 1–94 (2017). https://doi.org/10.1017/S0962492917000034

    Article  MathSciNet  MATH  Google Scholar 

  18. Güttel, S., van Beeumen, R., Meerbergen, K., Michiels, W.: NLEIGS: a class of fully rational Krylov methods for nonlinear eigenvalue problems. SIAM J. Sci. Comput. 36(6), A2842–A2864 (2014). https://doi.org/10.1137/130935045

    Article  MathSciNet  MATH  Google Scholar 

  19. Hammarling, S., Munro, C.J., Tisseur, F.: An algorithm for the complete solution of quadratic eigenvalue problems. ACM T. Math. Softw. 39(3), 1–19 (2013). https://doi.org/10.1145/2450153.2450156

    Article  MathSciNet  MATH  Google Scholar 

  20. Lanczos, C.: Trigonometric interpolation of empirical and analytical functions. J. Math. Phys. 17(1–4), 123–199 (1938). https://doi.org/10.1002/sapm1938171123

    Article  MATH  Google Scholar 

  21. Ledoux, V., Daele, M.V., Berghe, G.V.: Matslise: a MATLAB package for the numerical solution of Sturm-Liouville and schrödinger equations. ACM Trans. Math. Softw. (TOMS) 31(4), 532–554 (2005). https://doi.org/10.1145/1114268.1114273

    Article  MATH  Google Scholar 

  22. Malik, M., Huy, D.H.: Vibration analysis of continuous systems by differential transformation. Appl. Math. Comput. 96(1), 17–26 (1998). https://doi.org/10.1016/S0096-3003(97)10076-5

    MathSciNet  MATH  Google Scholar 

  23. Matos, J.M.A., Rodrigues, M.J., Matos, J.C.: Explicit formulae for integro-differential operational matrices. Math. Comput. Sci. 15, 45–61 (2021). https://doi.org/10.1007/s11786-020-00465-1

    Article  MathSciNet  MATH  Google Scholar 

  24. McFadden, G.B., Murray, B.T., Boisvert, R.F.: Elimination of spurious eigenvalues in the Chebyshev Tau spectral method. J. Comput. Phys. 91(1), 228–239 (1990). https://doi.org/10.1016/0021-9991(90)90012-P

    Article  MathSciNet  MATH  Google Scholar 

  25. Moler, C.B., Stewart, G.W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10(2), 241–256 (1973). https://doi.org/10.1137/0710024

    Article  MathSciNet  MATH  Google Scholar 

  26. Olver, S., Townsend, A.: A fast and well-conditioned spectral method. SIAM Rev. 55(3), 462–489 (2013). https://doi.org/10.1137/120865458

    Article  MathSciNet  MATH  Google Scholar 

  27. Orszag, S.A.: Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (04), 689 (1971). https://doi.org/10.1017/S0022112071002842

    Article  MATH  Google Scholar 

  28. Ortiz, E.L., Samara, H.: An operational approach to the Tau method for the numerical solution of non-linear differential equations. Computing 27, 15–26 (1981). https://doi.org/10.1007/BF02243435

    Article  MathSciNet  MATH  Google Scholar 

  29. Ortiz, E.L., Samara, H.: Numerical solution of differential eigenvalue problems with an operational approach to the Tau method. Computing 31(2), 95–103 (1983). https://doi.org/10.1007/BF02259906

    Article  MathSciNet  MATH  Google Scholar 

  30. Pruess, S., Fulton, C.T.: Mathematical software for Sturm-Liouville problems. ACM Trans. Math. Softw. (TOMS) 19(3), 360–376 (1993). https://doi.org/10.1145/155743.155791

    Article  MATH  Google Scholar 

  31. Pryce, J.D.: Error control of phase-function shooting methods for Sturm-Liouville problems. IMA J. Numer. Anal. 6(1), 103–123 (1986). https://doi.org/10.1093/imanum/6.1103

    Article  MathSciNet  MATH  Google Scholar 

  32. Pryce, J.D., Marletta, M.: A new multi-purpose software package for Schrödinger and Sturm-Liouville computations. Comput. Phys. Commun. 62(1), 42–52 (1991). https://doi.org/10.1016/0010-4655(91)90119-6

    Article  MATH  Google Scholar 

  33. Reddy, S.C., Henningson, D.S.: Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238 (1993). https://doi.org/10.1017/S0022112093003738

    Article  MathSciNet  MATH  Google Scholar 

  34. Reddy, S.C., Schmid, P.J., Henningson, D.S.: Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Math. 53(1), 15–47 (1993). https://doi.org/10.1137/0153002

    Article  MathSciNet  MATH  Google Scholar 

  35. Rommes, J.: Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems Ax = λBx with singular B. Math. Comput. 77 (262), 995–1016 (2007). https://doi.org/10.1090/s0025-5718-07-02040-6

    Article  MathSciNet  MATH  Google Scholar 

  36. Solov’ëv, S.I.: Preconditioned Iterative methods for a class of nonlinear eigenvalue problems. Linear Algebra Appl. 415(1), 210–229 (2006 ). https://doi.org/10.1016/j.laa.2005.03.034

    Article  MathSciNet  MATH  Google Scholar 

  37. Tisseur, F., Higham, N.J.: Structured pseudospectra for polynomial eigenvalue problems, with applications. SIAM J. Matrix Anal. Appl. 23(1), 187–208 (2001). https://doi.org/10.1137/S0895479800371451

    Article  MathSciNet  MATH  Google Scholar 

  38. Trindade, M., Matos, J., Vasconcelos, P.B.: Towards a Lanczos’ τ-method toolkit for differential problems. Math Comp. Sci. 10(3), 313–329 (2016). https://doi.org/10.1007/s11786-016-0269-x

    Article  MathSciNet  MATH  Google Scholar 

  39. Yuan, S., Ye, K., Xiao, C., Kennedy, D., Williams, F.: Solution of regular second-and fourth-order Sturm-Liouville problems by exact dynamic stiffness method analogy. J. Eng. Math. 86(1), 157–173 (2014). https://doi.org/10.1007/s10665-013-9646-5

    Article  MathSciNet  MATH  Google Scholar 

  40. Zebib, A.: Removal of spurious modes encountered in solving stability problems by spectral methods. J. Comput. Phys. 70(2), 521–525 (1987). https://doi.org/10.1016/0021-9991(87)90193-8

    Article  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous referees for their careful reading and valuable suggestions.

Funding

This work was partially supported by the Spanish Agencia Estatal de Investigación under grant PID2019-107379RB-I00 / AEI / 10.13039/501100011033, and by CMUP, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020.

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Authors and Affiliations

  1. CMUP, Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4299-464, Porto, Portugal

    P.B. Vasconcelos

  2. Universitat Politècnica de València, D. Sistemes Informàtics i Computació, Camí de Vera, s/n, E-46022, València, Spain

    J.E. Roman

  3. Centro de Matemática da Universidade do Porto e Instituto Superior de Engenharia, Rua Dr. António Bernardino de Almeida, 431,4249-015, Porto, Portugal

    J.M.A. Matos

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Vasconcelos, P., Roman, J. & Matos, J. Solving differential eigenproblems via the spectral Tau method. Numer Algor 92, 1789–1811 (2023). https://doi.org/10.1007/s11075-022-01366-z

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