an:04080806
Zbl 0661.65084
Gardner, David R.; Trogdon, Steven A.; Douglass, Rod W.
A modified tau spectral method that eliminates spurious eigenvalues
EN
J. Comput. Phys. 80, No. 1, 137-167 (1989).
00169384
1989
j
65L15 34L99
tau spectral method; spurious eigenvalues; truncated series expansion; numerical instability; Orr-Sommerfeld stability equation; Poiseuille flow; convergence
The tau spectral method proposed by \textit{C. Lanczos} [Applied Analysis (1956; Zbl 0074.105)] for solving eigenvalue problems in ordinary differential equations is modified so that spurious eigenvalues are eliminated. It uses a truncated series expansion of Chebyshev polynomials in a set of complete functions as an approximation for the solution of the differential equation.
The modification involves an appropriate factorization of the differential operator which removes the numerical instability. The modified tau method for a general fourth-order eigenvalue problem, for a system of fourth-order equations, and for the Orr-Sommerfeld stability equation for plane Poiseuille flow is discussed and compared with the ordinary tau method. The modified method converges at least as rapidly as the usual method. The use of the tau coefficients as identifiers of spurious eigenvalues and indicators of convergence is shown.
V.Burjan
Zbl 0074.105