In this paper we are concerned with numerical stability of Chebyshev-tau method in solving some singularly perturbed two-point boundary value problems.
We consider linear as well as nonlinear (convection-dominated flow) problems. In order to avoid the lack of numerical stability of this method we try a smoothing technique as well as a domain decomposition for linear problems.
Some successful numerical experiments are carried out.
Tiberiu Popoviciu Institute of Numerical Analysis
Babeș-Bolyai University, Faculty of Mathematics
Chebyshev-tau; stability; two-point boundary value problem; singularly perturbed; steady state Burger; smoothing;
See the expanding block below.
C. I. Gheorghiu, S.I. Pop, On the Chebyshev-tau approximation for some singularly perturbed two point boundary value problems – Numerical experiments, Rev. Anal. Numér. Théor. Approx. 24 (1995) 117-124.
Rev. Anal. Numér. Théor. Approx.
Editions de l’Academie Roumaine
Paper in html format
 C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A., Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, Springer Series in Computational Physics, 1988.
 C. I. Gheorghiu, On a Linear Singularity Perturbed TPBVP, Univ. of Cluj-Napoca, Seminar of Functional Analysis and Numerical Methods, Preprint 1 (1988), pp. 67-74.
 D. Gottlieb, S.A. Orszag, Numerical analysis of Spectral Methods: Theory and Applications, SIAM Philadelphia, 1977.
 C. Johson, Numerical solutions of p.d.e. by the f.e.m., Cambridge Univ. Press. 1987.
 R. B. Kellog, A. Tsan, analysis of some difference approximations for a singular perturbaiton problem without turning points, Math. Comput. 32, pp. 1025-1039 (1978).
 Y. Maday, A. Quarteroni, Legendre and Chebyshev Spectral Approixmation of Burgers’ Equation, Numer. Math. 37, pp. 321-332 (1981).
 S.A. Orszag, Accurate Solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50, pp. 689-703 (1971).
 M. Stynes, E. O’Riordan, An analysis of a t.p.b.c.p. with a boundary layer, using only finite element techniques, Univ. College Cork. Ireland, 1989.