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

作者：Meerbergen Karl^*; Schroeder Christian; Voss Heinrich

来源：Numerical Linear Algebra with Applications, 2013, 20(5): 852-868.

DOI：10.1002/nla.1848

• two-parameter eigenvalue problem
• Jacobi-Davidson
• nonlinear eigenvalue problem
• delay-differential equation
• critical delay

The 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.

• 出版日期2013-10