N.D. Aparicio, S.J.A. Malham, and M. Oliver,
Numerical evaluation of the Evans function by Magnus
integration,
BIT Numerical Mathematics 45 (2005), 219-258.
Abstract:
We use Magnus methods to compute the Evans function for spectral
problems as arise when determining the linear stability of travelling
wave solutions to reaction-diffusion and related partial differential
equations. In a typical application scenario, we need to repeatedly
sample the solution to a system of linear non-autonomous ordinary
differential equations for different values of one or more parameters
as we detect and locate the zeros of the Evans function in the right
half of the complex plane.
In this situation, a substantial portion of the computational effort -
the numerical evaluation of the iterated integrals which appear in the
Magnus series - can be performed independent of the parameters and
hence needs to be done only once. More importantly, for any given
tolerance Magnus integrators possess lower bounds on the step size
which are uniform across large regions of parameter space and which
can be estimated a priori. We demonstrate, analytically as
well as through numerical experiment, that these features render
Magnus integrators extremely robust and, depending on the regime of
interest, efficient in comparison with standard ODE solvers.
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