Software Development for Singular BVPs

In course of the project we studied the solution of problems known from various applications in physics, chemistry and environmental sciences. In the computer simulation of the run-up of avalanches one tries to specify the place and size of possible barriers placed in the landscape in order to stop the avalanche before it can reach especially designed regions (villages for instance). Mathematical models of such applications can be written in form of singular initial or boundary value problems. In general such models cannot be solved exactly - the only option is to provide a numerical approximation for the exact solution. The term "singularity" refers to certain unsmoothness in the problem data which makes it difficult to use standard software to solve such a problem. In general, the standard software applied to singular systems works inefficiently, or it does not work at all. Our goal was to provide a theoretical foundation for a program designed to solve singular problems in an efficient way. The following principles were to follow: 1. We know that the numerical solution is not exact and its error shall satisfy a tolerance requirement specified by the user. Clearly, to fulfill such a task we need to be able to control the size of the error. It seems natural to require that the error of the approximate solution shall decrease when the effort we invest to calculate the solution increases. Numerical methods having such a property are called convergent. With growing effort the values of the approximation approach the values of the exact solution. It is important to know how fast the error decreases. 2. Since we do not know the exact solution in general, we do not know the error of the approximation. Still we need a dependable information on its size, in order to be able to satisfy user`s tolerance requirement. Such an information will be provided by a correct and dependable error estimation procedure. Moreover, we use this estimate to solve the problem efficiently - in the region where the error is large, we will increase the amount of work, but we will avoid to do so in regions where the solution is accurate already. In a reasonably controlled program the effort increases only if the problem is becoming more difficult or if the user requires more accuracy. In course of the project we were able to propose numerical solution methods for large classes of singular problems and prove the fast convergence of these methods. Moreover, we were able to design a dependable and asymptotically correct error estimation procedure used as a basis for the efficient grid selection strategy. These routines were the basis for the software design. Two programs are available - for the numerical solution of boundary value problems of second order (based on finite difference approximations) and initial value problems (based on the so-called defect correction principle) of the first order. The second program will be the tool for the treatment of the models describing the run-up of avalanches mentioned above.