A Jacobi-Davidson method for two-real-parameter nonlinear eigenvalue problems arising from delay-differential equations

• Published in: Numerical linear algebra with applications 2013-10, Vol.20 (5), p.852-868
• Main Authors: Meerbergen, Karl, Schröder, Christian, Voss, Heinrich
• Format: Article
• Language: English
• Subjects: critical delay; Delay; delay-differential equation; Eigenvalues; Jacobi-Davidson; Linear algebra; Linear systems; Mathematical analysis; Mathematical models; nonlinear eigenvalue problem; Nonlinearity; Solvers; two-parameter eigenvalue problem
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.1848

SUMMARYThe critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.