Over the past decade we have developed two method-of-lines B-Spline collocation software packages, BACOL and BACOL, for the numerical solution of 1D parabolic PDEs. These packages have been shown to be efficient, reliable and robust, especially for problems with solutions exhibiting sharp layers, and for stringent tolerances. The packages feature adaptive control of estimates of the spatial and temporal errors. In BACOL the temporal integration and error control is handled by the BDF solver DASSL, while BACOLR employs the Runge-Kutta solver RADUA5.
While the BACOL/BACOLR spatial error estimates are generally quite reliable, the spatial error estimation algorithm involves the (expensive) computation of two collocation solutions of orders $p$ and $p+1$. (The higher order solution is used to provide a spatial error estimate for the lower order solution.) This talk will discuss recent work investigating more efficient spatial error estimation algorithms based on (i) an order $p+1$ (superconvergent) interpolant that allows us to avoid the computation of the higher order collocation solution, and (ii) an order $p$ interpolant, whose error agrees asymptotically with the error of the lower order collocation solution, that allows us to avoid the computation of the lower order collocation solution. We have implemented new, more efficient versions of BACOL and BACOLR based on these new error estimation schemes that we call BACOLI and BACOLRI. We provide numerical results comparing the original versions of BACOL and BACOLR with the new versions and show that the latter can be about twice as fast as the originals.