Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems

Applied Numerical Mathematics

Volumn 22, Issue 4, 1996, Pages 399-434

  Dongarra, J J ^a,b   Straughan, B ^b,c   Walker, D W ^b  

^a University of Tennessee Knoxville   (United States)

^b OAK RIDGE NATIONAL LABORATORY   (United States)

^c The University   (United Kingdom)

Author keywords

Chebyshev polynomials; Eigenvalue problems; Multilayer flows; Orr sommerfeld equations; Qz algorithm

Indexed keywords

CHEBYSHEV APPROXIMATION; DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; HYDRODYNAMICS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; PARAMETER ESTIMATION; PIPE FLOW; POLYNOMIALS; PROBLEM SOLVING; SPECTRUM ANALYSIS; VISCOUS FLOW;

CHEBYSHEV POLYNOMIALS; CHEBYSHEV TAU METHOD; COUETTE FLOW; EIGENVALUE PROBLEMS; HYDRODYNAMIC STABILITY PROBLEMS; MULTILAYER FLOWS; ORR SOMMERFELD EQUATIONS; POISEUILLE FLOW; QZ ALGORITHMS;

ALGORITHMS;

EID: 0030401362     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(96)00049-9     Document Type: Article

References (30)

1. A.A. Abdullah and K.A. Lindsay, Some remarks on the computation of the eigenvalues of linear systems, Math. Models Methods Appl. Sci. 1 (1991) 153-165.
2. E. Anderson, Z. Bai, C. Bischof, J. Demmel, J.J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov and D. Sorensen, LAPACK Users' Guide (SIAM, Philadelphia, PA, 2nd ed., 1995).
3. K.M. Butler and B.F. Farrell, Three-dimensional optimal perturbations in viscous shear flow, Phys. Fluids A 4 (1992) 1637-1650.
4. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer, Berlin, 1988).
5. K.P. Chen and D.G. Crighton, Instability of the large Reynolds number flow of a Newtonian fluid over a viscoelastic fluid, Phys. Fluids A 6 (1994) 152-163.
6. J. Choi, J.J. Dongarra, R. Pozo, D.C. Sorensen and D.W. Walker, CRPC research into linear algebra software for high performance computers. Internat. J. Supercomputer Appl. 8 (1994) 99-118.
7. A. Davey, A simple numerical method for solving Orr-Sommerfeld problems, Quart. J. Mech. Appl. Math. 26 (1973) 401-411.
8. J. Demmel and B. Kågström, The generalized Schur decomposition of an arbitrary pencil A - λB: robust software with error bounds and applications. Part I: theory and algorithms, ACM Trans. Math. Software 19 (1993)160-174.
9. J. Demmel and B. Kågström, The generalized Schur decomposition of an arbitrary pencil A - λB: robust software with error bounds and applications. Part II: software and applications, ACM Trans. Math. Software 19 (1993) 175-201.
10. J.J. Dongarra and D.W. Walker, Software libraries for linear algebra computations on high performance computers, SIAM Rev. 37 (1995) 151-180.
11. RG. Drazin and W.H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1981).
12. D.R. Fearn, Eigensolutions of boundary value problems using inverse iteration, J. Comput. Appl. Math. 34 (1991) 201-209.
13. L. Fox, Chebyshev methods for ordinary differential equations, Comput. J. 4 (1962) 318-331.
14. D.R. Gardner, S.A. Trogdon and R.W. Douglas, A modified tau spectral method that eliminates spurious eigenvalues, J. Comput. Phys. 80 (1989) 137-167.
15. D.B. Haidvogel and T Zang, The accurate solution of Poisson's equation by expansion in Chebyshev polynomials, J. Comput. Phys. 30 (1979) 167-180.
16. A.P. Hooper, The stability of two superposed viscous fluids in a channel, Phys. Fluids A 1 (1989) 1133-1142.
17. W. Huang and D.M. Sloan, The pseudospectral method for solving differential eigenvalue problems, J. Comput. Phys. 111 (1994) 399-409.
18. K.A. Lindsay and R.R. Ogden, A practical implementation of spectral methods resistant to the generation of spurious eigenvalues, Internat. J. Numer. Methods Fluids 15 (1992) 1277-1294.
19. G.B. McFadden, B.T. Murray and R.F. Boisvert, Elimination of spurious eigenvalues in the Chebyshev tau spectral method, J. Comput. Phys. 91 (1990) 228-239.
20. C.B. Moler and G.W. Stewart, An algorithm for generalized matrix eigenproblems, SIAM J. Numer. Anal. 10 (1973) 241-256.
21. G. Mulone, On the nonlinear stability of a fluid layer of a mixture heated and salted from below, Contin. Mech. Thermodyn. 6 (1994) 161-184.
22. S.A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech. 50 (1971) 689-703.
23. S.C. Reddy and D.S. Henningson, Energy growth in viscous channel flows, J. Fluid Mech. 252 (1993) 209-238.
24. Y. Renardy, Weakly nonlinear behaviour of periodic disturbances in two-layer Couette-Poiseuille flow, Phys. Fluids A 1 (1989) 1666-1676.
25. G.W. Stewart and J.-G. Sun, Matrix Perturbation Theory (Academic Press, New York, 1990).
26. B. Straughan and D.W. Walker, Anisotropic porous penetrative convection, Proc. Roy. Soc. London Ser. A 452 (1996) 97-115.
27. B. Straughan and D.W. Walker, Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems, J. Comput. Phys., to appear.
28. B. Straughan and D.W. Walker, Multi-component diffusion and penetrative convection, to appear.
29. C.S. Yih, Instability due to viscosity stratification, J. Fluid Mech. 27 (1967) 337-352.
30. A. Zebib, Removal of spurious modes encountered in solving stability problems by spectral methods, J. Comput. Phys. 70 (1987) 521-525.