scholarly journals A Subspace Iteration Algorithm for Fredholm Valued Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Christian Engström ◽  
Luka Grubišić
Keyword(s):  
Numerical Experiments ◽  
Eigenvalue Problems ◽  
Spectral Component ◽  
Contour Integration ◽  
Iteration Algorithm ◽  
Nonlinear Eigenvalue Problems ◽  
Infinite Dimensional ◽  
Subspace Iteration ◽  
Fredholm Theorem ◽  
Nonlinear Eigenvalue

We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algorithm for polynomial and rational eigenvalue problems.

Solution of Thermoacoustic Eigenvalue Problems With a Noniterative Method

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Philip E. Buschmann ◽  
Georg A. Mensah ◽  
Franck Nicoud ◽  
Jonas P. Moeck
Keyword(s):  
Eigenvalue Problems ◽  
Contour Integration ◽  
Complex Eigenvalue ◽  
Combustion Dynamics ◽  
Element Discretization ◽  
Nonlinear Eigenvalue Problems ◽  
Flame Response ◽  
The Stability ◽  
Thermoacoustic Oscillations ◽  
Nonlinear Eigenvalue

Abstract Gas turbine combustors are prone to undesirable combustion dynamics in the form of thermoacoustic oscillations. Analysis of the stability of thermoacoustic systems in the frequency domain leads to nonlinear eigenvalue problems (NLEVP); here, “nonlinear” refers to the fact that the eigenvalue, the complex oscillation frequency, appears in a nonlinear fashion. In this paper, we employ a noniterative strategy based on contour integration in the complex eigenvalue plane, which returns all eigenvalues inside the contour. An introduction to the technique is given, and is complemented with guidelines for the specific application to thermoacoustic problems. Two prototypical nonlinear eigenvalue problems are considered: a network model of the classical Rijke tube with an analytic flame response model and a finite element discretization of an annular model combustor with an experimental flame transfer function (FTF). Computation of all eigenvalues in a domain of interest is vital to assess stability of these systems. We demonstrate that this is generally challenging for iterative strategies. An eigenvalue solver based on contour integration, in contrast, provides a reliable, noniterative method to achieve this goal.

Solution of Thermoacoustic Eigenvalue Problems With a Non-Iterative Method

2019 ◽  
Author(s):  
Philip E. Buschmann ◽  
Georg A. Mensah ◽  
Franck Nicoud ◽  
Jonas P. Moeck
Keyword(s):  
Iterative Method ◽  
Eigenvalue Problems ◽  
Contour Integration ◽  
Complex Eigenvalue ◽  
Combustion Dynamics ◽  
Element Discretization ◽  
Nonlinear Eigenvalue Problems ◽  
Function Computation ◽  
Flame Response ◽  
Nonlinear Eigenvalue

Abstract Gas turbine combustors are prone to undesirable combustion dynamics in the form of thermoacoustic oscillations. Analysis of the stability of thermoacoustic systems in the frequency domain leads to nonlinear eigenvalue problems; here, ‘nonlinear’ refers to the fact that the eigenvalue, the complex oscillation frequency, appears in a nonlinear fashion. In this paper, we employ a non-iterative strategy based on contour integration in the complex eigenvalue plane, which returns all eigenvalues inside the contour. An introduction to the technique is given, and is complemented with guidelines for the specific application to thermoacoustic problems. Two prototypical nonlinear eigenvalue problems are considered: a network model of the classical Rijke tube with an analytic flame response model and a finite element discretization of an annular model combustor with an experimental flame transfer function. Computation of all eigenvalues in a domain of interest is vital to assess stability of these systems. We demonstrate that this is generally challenging for iterative strategies. An eigenvalue solver based on contour integration, in contrast, provides a reliable, non-iterative method to achieve this goal.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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