## Abstract

In this paper, we consider the numerical treatment of singular eigenvalue problems supplied with eigenparameter dependent boundary conditions using spectral methods. On the one hand, such boundary conditions hinder the construction of test and trial space functions which could incorporate them and thus providing well-conditioned Galerkin discretization matrices. On the other hand, they can generate surprising behavior of the eigenvectors hardly detected by analytic methods. These singular problems are often indirectly approximated by regular ones. We argue that spectral collocation as well as tau method offer remedies for the first two issues and provide direct and efficient treatment to such problems. On a finite domain, we consider the so-called Petterson-König’s rod eigenvalue problem and on the half line, we take into account the Charney’s baroclinic stability problem and the Fourier eigenvalue problem. One boundary condition in these problems depends on the eigenparameter and additionally, this also could depend on some physical parameters. The Chebyshev collocation based on both, square and rectangular differentiation and a Chebyshev tau method are used to discretize the first problem. All these schemes cast the problems into singular algebraic generalized eigenvalue ones which are solved by the QZ and/or Arnoldi algorithms as well as by some target oriented Jacobi-Davidson methods. Thus, the spurious eigenvalues are completely eliminated. The accuracy of square Chebyshev collocation is roughly estimated and its order of approximation with respect to the eigenvalue of interest is determined. For the problems defined on the half line, we make use of the Laguerre-Gauss-Radau collocation. The method proved to be reliable, accurate, and stable with respect to the order of approximation and the scaling parameter.

## Authors

Călin-Ioan **Gheorghiu**

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

## Keywords

Collocation; Chebyshev; Laguerre; square differentiation; rectangular differentiation; tau method; Petterson-Konig’s rod; Charney stability; singular eigenproblems.

### References

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## Cite this paper as

C.I. Gheorghiu, *On the numerical treatment of the eigenparameter dependent boundary conditions*, Numerical Algorithms, **77** (2018) no. 1, pp 77–93, DOI: 10.1007/s11075-017-0305-1

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## About this paper

##### Print ISSN

1017-1398

##### Online ISSN

1572-9265

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## References

[1] Boyd, J.P.: *Orthogonal rational functions on a semi-infinite interval*. J. Comp. Phys.70, 63–88 (1987)

[2] Boyd, J.P.: *Chebyshev domain truncation is inferior to Fourier domain truncation for solving problems on an infinite interval*. J. Sci. Comput. 3, 109–120 (1988)

[3] Boyd, J.P., Rangan, C., Bucksbaum, P.H.: *Pseudospectral methods on a semi-infinite interval with application to the hydrogen atom: a comparison of the mapped Fourier sine **method with Laguerre series and rational Chebyshev expansions*. J. Comp. Phys.188, 56–74 (2003)

[4] Boyd, J.P.: *Five themes in Chebyshev spectral methods applied to the regularized Charney eigen-problem: extra numerical boundary conditions, a boundary-layer-resolving change of
*

*coordinate, parametrzing a curve which is singularat at an endpoint, extending the tau method to log-and-polynomials and finding the roots of a polynomial-and-log approximation*. Comput. Math. Appl.71,1277–1241 (2016)

[5] Branscome, L.E.: *The charney baroclinic stability problem: approximate solutions and modal structures*. J. Atmos. Sci., 1393–1409 (1983)

[6] Driscoll, T.A., Hale, N., Trefethen, L.N.: *Chebfun Guide*. Pafnuty Publications, Oxford (2014)

[7] Driscoll, T.A., Hale, N.: *Rectangular spectral collocation*. IMA J. Numer. Anal. doi:10.1093/imanum/dru062

[8] Gheorghiu, C.I., Pop, I.S.: *A modified Chebyshev-Tau method for a hydrodynamic stability problem*. Approximation and Optimization. Proceedings of the International Conference on Approximation and Optimization (Romania)-ICAOR Cluj-Napoca, vol. II, pp. 119–126 (1996)

[9] Gheorghiu, C.I., Rommes, J.: *Application of the Jacobi-Davidson method to accurate analysis of singular linear hydrodynamic stability problems*. Int. J. Numer. Method Fl. (2012)

[10] Gheorghiu, C.I.: *Spectral methods for non-standard eigenvalue problems. Fluid and Structural Mechanics and Beyond*. Springer Cham Heidelberg New York Dondrecht London (2014)

[11] Gheorghiu, C.I.: *Stable spectral collocation solutions to a class of Benjamin Bona Mahony initialvalue problems*. Appl. Math. Comput. 273, 1090–1099 (2016)

[12] Gheorghiu, C.I.: *Spectral collocation solutions to systems of boundary layer type*. Numer. Algor. doi:10.1007/s11075-015-0083-6

[13] Gottlieb, D., Orszag, S.A.: *Numerical analysis of spectral methods: theory and applications*. SIAM,Philadelphia (1977)

[14] Goussis, D.A., Pearlstein, A.J.*: Removal of infinite eigenvalues in the generalized matrix eigenvalue problems*. J. Comput. Phys.84, 242–246 (1998)

[15] Maohzu, Z., Sun, J., Zettl, A.: *The spectrum of singular Sturm-Liouville problems with eigen-parameter dependent boundary conditions and its approximation*. Results. Math.63, 1311—1330(2013)

[16] van Noorden, T., Rommes, J.: *Computing a partial generalized real Schur form using the Jacobi-Davidson method*. Numer. Linear Algebra Appl.14, 197–215 (2007)

[17] Rommes, J.: *Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problemsAx=λBx with B singular*. Math. Comput. 77, 995–1015 (2008)

[18] Shen, J., Tang, T., Wang, L.-L.: *Spectral methods*. Algorithms, Analysis and Applications SpringerHeidelberg Dordrecht London New York (2011)

[19] Trefethen, L.N.: *Computation of pseudospectra*. Acta Numer.8, 247–295 (1999)

[20] Tretter, Ch.: *Nonselfadjoint spectral problems for linear pencils N−λP of ordinary differential operators with λ-linear boundary conditions: completeness results*. Integr. Equat. Oper. Th.26, 222–248(1996)

[21] Tretter, C.h.: *A linearization for a class of λ−nonlinear boundary eigenvalue problems*. J. Math.Analysis Appl.247, 331–355 (2000)

[22] Tretter, Ch.: *Boundary eigenvalue problems for differential equations Nη=λPη with λ-polynomial boundary conditions*. Integr. J. Diff. Eqns.170, 408–471 (2001)

[23] Xu, K., Hale, N.: *Explicit construction of rectangular differential matrices*. IMA J. Numer. Anal.doi:10.1093/imanum/drv013

[24] Xu, K., Hale, N.: *The Chebyshev points of the first kind*. Appl. Numer. Math.102, 17–30 (2016)

[25] Wang, C.M., Wang, C.Y., Reddy, J.N.: *Exact solutions for buckling of structural members*. CRC Press(2005)

[26] Weideman, J.A.C., Reddy, S.C.: *A MATLAB differentiation matrix suite*. ACM T. Math. Software26, 465–519 (2000)

[27] von Winckel, G.: *Fast Chebyshev Transform*, MathWorks. File Exchanges (2005)