scholarly journals Gegenbauer Tau Methods With and Without Spurious Eigenvalues

2009 ◽  
Vol 47 (1) ◽  
pp. 48-68 ◽  
Author(s):  
Marios Charalambides ◽  
Fabian Waleffe
Keyword(s):  
Spurious Eigenvalues

On the disturbed motion of a plane vortex sheet

1958 ◽  
Vol 4 (5) ◽  
pp. 538-552 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Approximation ◽  
Formal Solution ◽  
Vortex Sheet ◽  
Inviscid Fluid ◽  
Initial Value ◽  
Spatially Periodic ◽  
The Stability ◽  
Spurious Eigenvalues ◽  
Disturbed Motion

A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity.

THE TROUBLE WITH SPURIOUS EIGENVALUES

2007 ◽  
Vol 17 (04) ◽  
pp. 1375-1381 ◽  
Author(s):  
RAYMOND A. ADOMAITIS
Keyword(s):  
Stability Analysis ◽  
Numerical Analysis ◽  
Eigenvalue Problem ◽  
Mean Value ◽  
Galerkin Projection ◽  
Invariant Circle ◽  
Eigenvalues And Eigenfunctions ◽  
The Mean ◽  
Spurious Eigenvalues

Stability analysis of spatially discretized approximations to invariant circle solutions to a noninvertible map is shown to produce seemingly spurious eigenvalues, found to exist for all truncation numbers. Numerical analysis of the spectral Galerkin projection procedure used to compute the solutions and the eigenvalues of the linearization shows that the set of discrete eigenvalues converges in terms of its mean value. Further analysis shows that the spurious eigenvalues — those that are farthest from the mean value — correspond to eigenfunctions which produce the largest residual in the eigenvalue problem used to compute the eigenvalues and eigenfunctions.

True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Jeng-Tzong Chen ◽  
Jia-Wei Lee ◽  
Ying-Te Lee ◽  
Wen-Che Lee
Keyword(s):  
Dirichlet Problem ◽  
Influence Function ◽  
Single Layer ◽  
Boundary Integral ◽  
Double Layer Potential ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Potential Formulation ◽  
The Neumann Problem ◽  
Spurious Eigenvalues

In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions.

scholarly journals A remark on spurious eigenvalues in a square

1999 ◽  
Vol 12 (3) ◽  
pp. 107-114 ◽  
Author(s):  
D. Boffi ◽  
R.G. Duran ◽  
L. Gastaldi
Keyword(s):  
Spurious Eigenvalues

On the null and nonzero fields for true and spurious eigenvalues of annular and confocal elliptical membranes

2013 ◽  
Vol 37 (1) ◽  
pp. 42-59 ◽  
Author(s):  
Jeng-Tzong Chen ◽  
Jia-Wei Lee ◽  
I.-Lin Chen ◽  
Po-Shen Kuo
Keyword(s):  
Spurious Eigenvalues
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