Spectral methods in linear stability. Applications to thermal convection with variable gravity field


The onset of convection in a horizontal layer of fluid heated from below in the presence of a gravity field varying across the layer is numerically investigated. The eigenvalue problem governing the linear stability of the mechanical equilibria of the fluid, in the case of free boundaries, is a sixth order differential equation with Dirichlet and hinged boundary conditions. It is transformed into a system of second order differential equations supplied only with Dirichlet boundary conditions. Then it is solved using two distinct classes of spectral methods namely, weighted residuals (Galerkin type) methods and a collocation (pseudospectral) method, both based on Chebyshev polynomials. The methods provide a fairly accurate approximation of the lower part of the spectrum without any scale resolution restriction. The Viola’s eigenvalue problem is considered as a benchmark one. A conjecture is stated for the first eigenvalue of this problem.


Călin-Ioan Gheorghiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Florica-Ioana Dragomirescu


Bénard convection; Variable gravity field; Hydrodynamic stability; High order two-point boundary value problem; Hinged boundary conditions; Spectral methods


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C.I. Gheorghiu, F.-I. Dragomirescu, Spectral methods in linear stability. Applications in thermal convection with variable gravity field, Appl. Numer. Math., 59 (2009) 1290-1302
doi: 10.1016/j.apnum.2008.07.004



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[1] M.G. Blyth, C. Pozrikidis, Effect of surfactant on the stability of film flow down an inclined plane, J. Fluid Mech. 521 (2004) 241–250.
[2] J.P. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover, New York, 2000.
[3] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, 1987.
[4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, 1961.
[5] J.J. Dongarra, B. Straughan, D.W. Walker, Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems, Appl. Numer. Math. 22 (1996) 399–434.
[6] I. Dragomirescu, Approximate neutral surface of a convection problem for variable gravity field, Rend. Sem. Mat. Univ. Politec Torino 64 (2006) 331–342.
[7] I. Dragomirescu, A. Georgescu, Linear stability bounds in a convection problem for variable gravity field, Bul. Acad. Stiinte Repub. Mold. Mat. 52 (2006) 51–66.
[8] I. Dragomirescu, A SLP-based method for a convection problem for a variable gravity field, in: Proc. of Aplimat 2007, Bratislava, pp. 149–154.
[9] I. Dragomirescu, Shifted polynomials in a convection problem, arXiv:0709.2240v1 [math-ph], 2007.
[10] P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, London, 1981.
[11] C.I. Gheorghiu, I.S. Pop, A modified Chebyshev-tau method for a hydrodynamic stability problem, in: Proc. ICAOR ’97, vol. II, Transylvania Press, Cluj-Napoca, 1997, pp. 119–126.
[12] C.I. Gheorghiu, Spectral Methods for Differential Problems, Casa Cartii de Stiinta Publishing House, Cluj-Napoca, 2007.
[13] L. Greenberg, M. Marleta, Numerical solution of non-self-adjoint Sturm–Liouville problems and related systems, SIAM J. Numer. Anal. 38  (2001) 1800–1845.
[14] W. Heinrichs, Improved condition number for spectral methods, Math. Comput. 53 (1998) 103–119.
[15] I. Herron, On the principle of exchange of stabilities in Rayleigh–Bénard convection, SIAM J. Appl. Math. 61 (2000) 1362–1368.
[16] A.A. Hill, B. Straughan, A Legendre spectral element method for eigenvalues in hydromagnetic stability, J. Comput. Appl. Math. 193 (2003) 363–381.
[17] W. Huang, D.M. Sloan, The pseudospectral methods for third-order differential equations, SIAM J. Numer. Anal. 29 (1992) 1626–1647.
[18] J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman & Hall, 2003.
[19] J.M. Melenk, N.P. Kirkner, V. Schwab, Spectral Galerkin discretization for hydrodynamic stability problems, Computing 65 (2000) 97–118.
[20] S. Orszag, Accurate solutions of Orr–Sommerfeld stability equation, J. Fluid Mech. 50 (1971) 689–703.
[21] K. Prakash, H. Raj, Effect of variable gravitational field on thermal instability of a rotating fluid layer with magnetic field in porous medium, Czech. J. Phys. 47 (1997) 793–800.
[22] S. Pruess, C.T. Fulton, Mathematical software for Sturm–Liouville problems, ACM Trans. Softw. 19 (1998) 360–376.
[23] A. Quarteroni, F. Saleri, Scientific Computing with MATLAB and Octave, second ed., Springer-Verlag, 2006.
[24] S.C. Reddy, P.J. Schmid, D. Henningson, Pseudospectra of Orr–Sommerfeld operator, SIAM J. Appl. Math. 53 (1993) 15–47.
[25] J. Shen, Efficient spectral-Galerkin method II, Direct solvers of second and fourth order equations by using Chebyshev polynomials, SIAM J. Sci. Comput. 16 (1995) 74–87.
[26] G.W. Stewart, J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.
[27] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer, Berlin, 2003.
[28] L.N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, PA, 2000.
[29] P.G. Vekilov, Protein crystal growth—microgravity aspects, Advances in Space Research 24 (1999) 1231–1240.
[30] J.A.C. Weideman, S.C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. on Math. Software 26 (2000) 465–519.

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