## Abstract

The aim of this paper is to show that the Jacobi–Davidson (JD) method is an accurate and robust method for solving large generalized algebraic eigenvalue problems with a singular second matrix. Such problems are routinely encountered in linear hydrodynamic stability analysis of flows that arise in various areas of continuum mechanics. As we use the Chebyshev collocation as a discretization method, the first matrix in the pencil is nonsymmetric, full rank, and ill‐conditioned. Because of the singularity of the second matrix, QZ and Arnoldi‐type algorithms may produce spurious eigenvalues. As a systematic remedy of this situation, we use two JD methods, corresponding to real and complex situations, to compute specific parts of the spectrum of the eigenvalue problems. Both methods overcome potentially severe problems associated with spurious unstable eigenvalues and are fairly stable with respect to the order of discretization. The real JD outperforms the shift‐and‐invert Arnoldi method with respect to the CPU time for large discretizations. Three specific flows are analyzed to advocate our statements, namely a multicomponent convection–diffusion in a porous medium, a thermal convection in a variable gravity field, and the so‐called Hadley flow.

## Authors

C.I. **Gheorghiu**

-Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

J. **Rommes
**NXP Semiconductors, Eindhoven, The Netherlands

## Keywords

## Cite this paper as:

C.I. Gheorghiu, J. Rommes, Application of the Jacobi-Davidson method to accurate analysis of singular linear hydrodynamic stability problems. Int. J. Numer. Meth. Fluids, 71 (2013) 358-369.

### References

see the expansion block below.

Not available yet.

## About this paper

##### Journal

Int. J. Numer. Meth. Fluids

##### Publisher Name

##### DOI

##### Print ISSN

?

##### Online ISSN

?

## MR

?

## ZBL

?

- Straughan B. Convection in a variable gravity field. Journal of Mathematical Analysis and Applications 1989; 140:467–475.
- Straughan B, Walker DW. Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. Journal of Computational Physics 1996; 127:128–141.
- Dondarra JJ, Straughan B, Walker DK. Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems. Applied Numerical Methods 1996; 22:399–434.
- Hill AA, Straughan B. A Legendre spectral element method for eigenvalues in hydrodinamic stability. Mathematical Methods in Applied Sciences 2006; 29:363–381.
- Hill AA, Straughan B. Linear and nonlinear stability tresholds for thermal convection in a box. Mathematical Methods in Applied Sciences 2006; 29:2123–2132.
- Valerio JV, Carvalho MS, Tomei C. Filtering the eigenvalues at infinity from the linear stability analysis of incompressible flows. Journal of Computational Physics 2007; 227:229–243.
- Boomkamp PAM, Boersma BJ, Miesen RHM, Beijnon GVA. A Chebyshev collocation method for solving two-phase flow stability problems. Journal of Computational Physics 1997; 132:191–200.
- Giannakis D, Fischer PF, Rosner R. A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free surface MHD. Journal of Computational Physics 2009; 228:1188–1233. DOI: 10.1016/j.jcp.2008.10.016.
- Khorrami MR, Malik MR, Ash LR. Application of spectral collocation techniques to the stability of swirling flows. Journal of Computational Physics 1989; 81:206–229.
- Gheorghiu CI, Dragomirescu IF. Spectral methods in linear stability. Application to thermal convection with variable gravity field. Applied Nunerical Mathematics 2009; 59:1290–1302. DOI: 10.1016/j.apnum.2008.07.004.
- Dragomirescu IF, Gheorghiu CI. Analitical and numerical solutions to an electrohydrodynamic stability problem. Applied Mathematics and Computation; 2010(216):3718–3727. DOI: 10.1016/j.acm.2010.05.028.
- Saad Y. Iterative Methods for Sparse Linear Systems. PWS Publishing: NY, 1996.
- Schmid PJ, Henningson DS. Stability and Transition in Shear Flows. Springer-Verlag: New York, 2001. 489.
- Zebib A. Removal of spurious modes encountered in solving stability problems by spectral methods. Journal of Computational Physics 1987; 70:521–525.
- Lindsay KA, Ogden RR. A practical implementation of spectral methods resistant to the generation of spurious eigenvalues. International Journal for Numerical Methods in Fluids 1992; 15:1277–1292.
- Melenk JM, Kirchner NP, Schwab C. Spectral Galerkin discretization for hydrodynamic stability problems. Computing 2000; 65:97–118.
- van Noorden TL, Rommes J. Computing a partial generalized real Schur form using the Jacobi–Davidson method. Numerical Linear Algebra with Applications 2007; 14:197–215.
- Sleijpen GL, van der Vorst HAA. A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM Journal of Matrix Analysis and Applications 1996; 17:401–425.
- Rommes J. Arnoldi and Jacobi–Davidson methods for generalized eigenvalue problems Ax D Bx with B singular. Mathematics of Computation 2008; 77:995–1015.
- Golub GH, Van der Vorst HA. Eigenvalue computation in the 20th century. Journal of Computational and Applied Mathematics 2000; 123:35–65.
- Valdettaro L, Rieutord M, Braconnier T, Frayssé V. Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi–Chebyshev algorithm. Journal of Computational and Applied Mathematics 2007; 205:382–393.
- Weideman JAC, Reddy SC. A MATLAB differentiation matrix suite. ACM Transactions in Mathematical Software 2000; 26:465–519.
- Golub GH, van Loan CF. Matrix Computations, third ed. The John Hopkins University Press: Baltimore, 1996.
- Fokkema DR, Sleijpen GLG, van der Vorst HA. Jacobi–Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM Journal of Scientific Computing 1998; 20:94–125.
- Sleijpen GLG, Booten JGL, Fokkema DR, van der Vorst HA. Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 1996; 36:595–633.
- Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal of Scientific and Statistic Computing 1986; 7:856–869.
- Meerbergen K, Spence A. Implicitely restarted Arnoldi with purification for the shift-invert transformation. Mathematics of Computation 1997; 66:667–689.
- Arnoldi WE. The principle of minimized iteration in the solution of the matrix eigenproblem. Quartely of Applied Mathematics 1951; 9:17–29.
- Bai Z, Demmel J, Dongarra J, Ruhe A, van der Vorst HA. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM: Philadelphia, 2000.
- van Dorsslaer JLM. Pseudospectra for matrix pencils and stability of equilibria. BIT 1997; 37:833–845.

soon