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Evaluating the Evans function: Order reduction in numerical methods

Authors: Simon Malham and Jitse Niesen
Journal: Math. Comp. 77 (2008), 159-179
MSC (2000): Primary 65L15; Secondary 65L20, 65N25
Published electronically: July 26, 2007
MathSciNet review: 2353947
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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.

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Additional Information

Simon Malham
Affiliation: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

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

Keywords: Evans function, Magnus method, order reduction.
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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