Recent Advances in Error Control B-spline Gaussian Collocation Software for PDEs

In this chapter we briefly review recent advances in Error Control B-spline Gaussian Collocation software for the numerical solution of 1D parabolic partial differential equations (PDEs). BACOL and BACOLR, two packages of this type, developed over the last decade, have been shown to be efficient, reliable, and robust, especially for problems having solutions with sharp moving layers and for stringent tolerances. These packages use high order methods in time and space and feature adaptive control of high order estimates of the temporal and spatial errors. The spatial error estimates require the computation of a second collocation solution, which introduces a substantial computational overhead. In order to address this issue, a new software package, called BACOLI, has recently been developed (through a substantial modification of BACOL) in which the computation of the second collocation solution is replaced by the computation of a high order interpolant. Numerical results have shown that BACOLI computes spatial error estimates that are generally of comparable quality to those computed by BACOL and that the new code is generally substantially more efficient than BACOL.