  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 Schr\"odinger
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.
