Evaluating the Evans function: Order reduction in numerical methods
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- by Simon Malham and Jitse Niesen PDF
- Math. Comp. 77 (2008), 159-179 Request permission
Abstract:
We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss–Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schrödinger operator. We show that although the fourth-order Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourth-order convergence in the end result is preserved. The Gauss–Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.References
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Additional Information
- Simon Malham
- Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
- Email: simonm@ma.hw.ac.uk
- Jitse Niesen
- Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
- Address at time of publication: Mathematics Department, La Trobe University, Victoria 3086, Australia
- Email: j.niesen@latrobe.edu.au
- Received by editor(s): April 20, 2006
- Received by editor(s) in revised form: November 15, 2006
- Published electronically: July 26, 2007
- Additional Notes: This work was supported by EPSRC First Grant GR/S22134/01.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 159-179
- MSC (2000): Primary 65L15; Secondary 65L20, 65N25
- DOI: https://doi.org/10.1090/S0025-5718-07-02016-9
- MathSciNet review: 2353947