When the Laplace transform is applied to a semi-discrete parabolic PDE, the result is a
contour integral of Bromwich type that has to be computed numerically. The integrand of
this formula involves the computation of the resolvent of a typically large matrix A, which
can be expensive. Therefore, for the sake of computational efficiency the number of function
evaluations has to be limited. This can be done via a judicious choice of quadrature scheme
and a matching good contour. The choice of the trapezoidal rule combined with a Hankel
contour has stood the test of time and forms the focal point of this talk.
Starting with the original proposal of A. Talbot in the mid-1970s (which was based on
the work of his student J. Green in the mid-1950s), we survey a number of such contours
that have been proposed in the literature. These contours are typically defined by a number
of parameters that can be tuned for optimal accuracy. Assuming some information on the
spectrum of A, we demonstrate how to find practical estimates for these parameters. A
consequence of this parameter tuning is that the error in the trapezoidal rule can be shown
to decrease exponentially with the number of nodes in the rule (also known as geometric
or spectral convergence). Thus it often happens that convergence to ten-digit accuracy or
better can be achieved with as few as a dozen nodes in the trapezoidal rule.
Additional issues that will be addressed in the talk are the effect of non-normality
in the matrix A (for example when solving a convection dominated problem), numerical
instability and the control of roundoff error, and the efficient computation of the resolvent
via recent techniques for the solution of shifted linear systems. Applications will be drawn
from the area of mathematical finance (notably the work on the Black-Scholes and Heston
equations done in collaboration with K. in ’t Hout of the University of Antwerp, Belgium).

[invited by J.-P. Berrut]