Defect Correction Techniques for Stiff ODEs

The application of acceleration techniqes for the numerical solution of stiff differential equations appears to be not unproblematic, since - due to the unsmoothness of the data functions defining a stiff ODE - the structure of discretization errors is quite complicated; pure asymptotic error expansions (as for smooth problems) do usually not exist in the stiff case. Indeed, as has been known for some time ago, extrapolation techniques fail in `mildly stiff` situations. Classical versions of defect correction algorithms turn out to be more robust, but also have certain drawbacks: No variants of defect correction algorithms have been known so far which achieve superconvergence (as it would e.g. be the case for a defect correction algorithm proposed by K.H.Schild applied to smooth boundary value problems). At the Institute for Applied Mathematics and Numerical Analysis, modified versions of the defect correction procedure have recently be proposed featuring convergence towards superconvergent fixed points - even for stiff problems with a very general geometry (more general compared with problems in standard singular perturbation form). The underlying algorithmic modifications are the following: (i) The use of an appropriately interpolated or locally integrated defect, (ii) for difficult stiff problems with moving stiff eigendirections: the use of a new, adaptive and algorithmically efficient realizable defect representation. Existing numerical experience shows that this is a very successful approach. In the project proposed we wish to pursue these isses further: (i) By means of systematic numerical experiments the performance of the various algorithmic variants shall be further tested and clarified, (ii) It will be attempted to obtain a deeper theoretical insight by means of analyzing appropriately selected stiff models.

This project was devoted to the investigation of defect correction algorithms for the numerical solution of differential equations. These are based on a correction via the residual (or defect) of a given approximation w.r.t. the given problem. But there are several ways to compute this defect, which may yield significantly different results, in particular when non-equidistant grids are involved or when stiff problems are considered. In this project a number of variants were implemented and numerically evaluated, in particular for stiff problems with varying degree of difficulty. Furthermore, the convergence of the so-called methods of defect quadrature and defect interpolation was successfully analyzed for smooth problems. In the stiff case the analysis is very expensive and was restricted to a scalar model problem. Here, similar to A-stability, domains of contraction of the defect correction variants were specified. The work is also related to project P 15072-N05, where similar techniques were used for the construction of a- posteriori error estimates for collocation methods (for singular problems, in particular).