The high quality COLSYS/COLNEW collocation software package is widely used for the numerical solution of boundary value ODEs (BVODEs), often through interfaces to computing environments such as Scilab, R, and Python. The continuous collocation solution returned by the code is much more accurate at a set of mesh points that partition the problem domain than it is elsewhere; the mesh point values are said to be superconvergent. In order to improve the accuracy of the continuous solution approximation at non-mesh points, when the BVODE is expressed in first order system form, an approach based on continuous Runge-Kutta (CRK) methods has been used to obtain a superconvergent interpolant (SCI) across the problem domain. Based on this approach, recent work has seen the development of a new, more efficient version of COLSYS/COLNEW that returns an error controlled SCI.

However, most systems of BVODEs include higher derivatives and a feature of COLSYS/COLNEW is that it can directly treat such mixed order BVODE systems, resulting in improved efficiency, continuity of the approximate solution, and user convenience. In this paper we generalize the approach mentioned above for first order systems to obtain SCIs for collocation solutions of mixed order BVODE systems. The main contribution of this paper is the derivation of generalizations of continuous Runge-Kutta-Nyström methods that form the basis for SCIs for this more general problem class. We provide numerical results that (ⅰ) show that the SCIs are much more accurate than the collocation solutions at non-mesh points, (ⅱ) verify the order of accuracy of these SCIs, and (ⅲ) show that the cost of utilizing the SCIs is a small fraction of the cost of computing the collocation solution upon which they are based.

This paper introduces BACOLIKR, a new software package for the error-controlled numerical solution of systems of one-dimensional time-dependent partial differential equations (PDEs). A novel feature of this package is that it allows the user to specify a solution dependent condition, called an event, and then the software will determine the point in time at which the specified event occurs. This event detection capability can be used to provide an efficient and accurate means for dealing with time-dependent discontinuities in the PDEs or the boundary conditions.

BACOLIKR employs adaptive B-spline Gaussian collocation for the spatial discretization of the PDE system within a spatial error control algorithm. The event detection capability in BACOLIKR is based on its use of a modified version of the time integrator, DASKR, which implements event detection for time-dependent differential-algebraic equations. BACOLIKR was developed through modifications of an earlier error control PDE solver, BACOLI.

In this paper, we provide an overview of the BACOLI and DASKR packages and then describe the modifications that were made in order to develop BACOLIKR. We then show how BACOLIKR can be used for the effective solution of a number of PDE-based event detection problems including solution layer-boundary intersection detection and solution layer merge detection in a fluid mechanics model, critical tumor mass detection in a brain tumor model, steady state detection in the Gierer-Meinhardt model, and boundary condition event detection in a discontinuous heat flow model.