Numerical Analysis |
Not very surprisingly, I am also interested in the numerical solution of differential equations, although I have given this domain less priority in recent years. I am mainly concerned with two different aspects: the treatment of general systems of differential equations - in this context often called DAE (differential algebraic equation) or PDAE (partial differential algebraic equation), resp. - and the numerical integration of Hamiltonian systems (with constraints). An overview over some of my ideas in this field is given in the article [54] . The first topic is closely related to my work on the formal theory of differential equations. In the papers [15, 45] I have shown that the formal theory (or more precisely, the completion to involution) provides a natural approach to define what is called the index of a (P)DAE. In contrast to the usual definition, ordinary and partial differential equations can thus be treated identically. As far as I know, this represents the first and still only definition of indices for arbitrary PDAE. In fact, there exist many different index concepts. The easiest are the differentiation indices which just count the number of completion steps you need until the system has certain properties. The relation of these to the perturbation index defined via an estimate for the solution of a perturbed equation was studied in [51] . A point that is probably a surprise for many numerical analysts (and it was a surprise for me, too) is that not just formal integrability but indeed involution is of importance here. This is demonstrated in the article [18] where the semi-discretisation of a linear partial differential system with constant coefficients is studied. I show that obstructions to involution may turn into integrability conditions upon semi-discretisation so that even if the original system is formally integrable but not involutive, the DAE obtained by the semi-discretisation has hidden integrability conditions. Concerning the second topic, I have tried to apply some of the results on constrained systems obtained by physicists to the problem of numerical integration. Up to now, this has been mainly the Dirac theory. In [16] I have studied the numerical properties of the Hamilton-Dirac equations, the equations of motion derived with the Dirac bracket. I have shown that (under rather modest conditions on the constraints) the constraint manifold is orbitally stable, so that only a slow drift appears. A problem of this approach is that one modifies here the symplectic structure of the phase space, so that standard symplectic integrators cannot be used. How severe this problem really is, still remains to be investigated. For most applications the actual use of the Hamilton-Dirac equations might not be very efficient, as it leads to the addition of complicated terms vanishing on the constraint manifold. If one uses projection methods, these terms do not contribute at all. But the Dirac theory can now be exploited to study theoretically the effect of the projections, as I did in the paper [13] . The results clearly demonstrate that position projections can have very bad effects and that in general momentum projections are not only cheaper but much more effective. Some further comments on the numerical analysis of constrained Hamiltonian systems are contained in [58] . For quite some time I have been working on a paper [89] on the impetus-striction formalism; unfortunately, I do not seem to find the time to finally finish it ... This formalism is closely related to the ideas mentioned above, as it is based on including momentum projections into the equations of motion. I have developed a purely Hamiltonian point of view of this approach which clarifies a number of points. |
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