Improved contour integral methods for parabolic PDEs

One way of computing the matrix exponential that arises in semidiscrete parabolic partial differential equations is via the Dunford-Cauchy integral formula. The integral is approximated by the trapezoidal or midpoint rules on a Hankel contour defined by a suitable change of variables. In a recent paper by Weideman & Trefethen (2007, Math. Comput., 76, 1341-1356) two widely used contours were analysed. Estimates for the optimal parameters that define these contours were proposed. In this paper this analysis is extended in two directions. First, the effect of roundoff error is now included in the error model. Second, we extend the results to the case of a model convection-diffusion equation, where a large convective term causes the matrix to be highly non-normal.