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Numerical integration of implicit delay differential equations and accurate computation of breaking points

Systems of implicit delay differential equations, including state-dependent problems, neutral and differential-algebraic equations, singularly perturbed problems, and small or vanishing delays are considered. The numerical integration of such problems is very sensitive to jump discontinuities in the solution or in its derivatives (so-called breaking points).
In this talk we discuss a new strategy - peculiar to implicit schemes - that allows codes to detect automatically and then to compute very accurately those breaking points which have to be inserted into the mesh to guarantee the required accuracy. In particular for state-dependent delays, where breaking points are not known in advance, this treatment leads to a significant improvement in accuracy. As a by-product we design strategies that are able to detect points of non-uniqueness or non-existence of the solution so that the code can terminate when such a situation occurs.
This is a joint work with Ernst Hairer (Geneva).

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