Software Development for Singular BVPs

The main goal of the project is the design of a code for the efficient and reliable solution of singular boundary value problems for systems of ordinary differential equations. Such problems occur in mathematical models of numerous applications from physics and chemistry, but also in ecology and material science. One of the ecology models deals with the simulation of the run-up and run-out of dry-flowing avalanches, where a leading-edge model is used to describe the avalanche`s dynamics. Singular problems also arise in dynamical systems when the arclength is used for the parametrization of the orbit. The search for a numerical method to be used as a basis in a code taking into account the specific difficulties caused by the singularity is strongly motivated by these applications and research activities in related areas. We propose to use collocation for the numerical solution of the underlying boundary value problem. This choice is motivated by its advantageous convergence properties, whereas in presence of singularity other direct higher order methods (finite differences) show order reductions and become inefficient. Clearly, an efficient code needs to be equipped with an asymptotically correct error estimation routine serving as a basis for the grid selection strategy. In our code we plan to control the global error instead of monitoring the local error. The main reason for this decision is the unsmoothness of the local error near the singular point and order reductions it often suffers from. Consequently, it turns out that grids generated via the equidistribution of the local error are usually very fine close to the singularity even when the solution is very smooth there. The experiments show that the equidistribution of the global error results in grids reflecting in a very proper way the solution behavior, remaining unaffected by the unsmoothness of the direction field close to the singularity. Usually, the estimation of the global error is expensive and therefore, the cost of the error estimate is an important issue. After certain modifications a method proposed by Stetter in 1978 and based on an idea due to Zadunaisky (1976) enables us to provide a reasonably cheap and asymptotically correct error estimate which seems to be most suitable for our purpose.