Kaspar Nipp

Invariant curves for variable step size integrators

AbstractThe behaviour of one-step methods with variable step size applied to

An extension of Tikhonov's theorem in singular perturbations for the planar case

\dot x = f(x)

CSE Program at ETH Zurich: Are We Doing the Right Thing?

is investigated. Usually the step size for the current step depends on one or several previous steps. However, under some natural assumptions it can be shown that the step size asymptotically depends only on the locationx. This allows to introduce anx-dependent time transformation taking a variable step size method to a constant step-size method. By means of such a transformation general properties of constant step size methods carry over to variable step size methods. This is used to show that if the differential equation admits a hyperbolic periodic solution the variable step size method admits an invariant closed curve near the orbit of the periodic solution.

Adapting the CSE Program at ETH Zurich to the Bologna Process

For singularly perturbed systems of ODEs satisfying a certain stability assumption the existence of an asymptotically stable (unstable) invariant manifold is proved. This invariant manifold is ε-close to the so-called reduced manifold of such a system. As an illustrative example a 3-dimensional autonomous system describing a model in biochemistry is considered.ZusammenfassungFür singulär gestörte Differentialgleichungssysteme, die einer gewissen Stabilitätsbedingung genügen, wird die Existenz einer asymptotisch stabilen (instabilen) invarianten Mannigfaltigkeit nachgewiesen, die in einer ε-Umgebung der sogenannten reduzierten Mannigfaltigkeit eines solchen Systems liegt. Als Anschauungsbeispiel wird ein dreidimensionales autonomes System betrachtet, welches ein biochemisches Modell beschreibt.

Graph Transform and Blow‐up in Singular Perturbations

A classical error estimate for singularly perturbed ordinary differential equations due to A. N. Tikhonov is generalized in the neighbourhood of a so-called leaving point where a certain stability assumption ceases to be valid. This is done for the planar case, and an application to the Van der Pol relaxation oscillator is given.ZusammenfassungEin klassisches Abschätzungsresultat der singulären Störungstheorie bei gewöhnlichen Differentialgleichungen geht auf A. N. Tikhonov zurück. Dieses Resultat wird verallgemeinert im ebenen Fall in der Nähe eines sogenannten „leaving point“, wo eine gewisse Stabilitätseigenschaft nicht mehr gilt. Zudem wird eine Anwendung auf den Relaxationsoszillator von Van der Pol gegeben.

Invariant Manifolds in Discrete and Continuous Dynamical Systems

SummaryWe present a numerical approach for solving in an efficient way stiff systems of singular perturbation type. This approach makes direct use of the theory of singularly perturbed IVPs. We determine numerically an expansion of an invariant manifold with respect to the small (stiffness) parameterɛ and we solve a reduced system of lower dimension no longer stiff on this manifold. The global error of the numerical approximation obtained contains anɛ-dependent term which decreases withɛ. This implies that the error does not increase as the stiffness of the system increases. We also suggest an appropriate implementation of the method and we give an application to some non-trivial example.ZusammenfassungEs wird ein numerisches Verfahren vorgeschlagen, das steife Systeme vom singulären Störungstyp auf effiziente Weise löst. Die Herleitung beruht auf analytischen Resultaten über singular gestörte Anfangswertprobleme. Eine invariante Mannigfaltigkeit wird numerisch entwickelt bezüglich des kleinen (Steifheits-) Parametersɛ und ein Differentialgleichungssystem kleinerer Dimension gelöst auf der Mannigfaltigkeit. Dieses reduzierte System ist zudem i. a. nicht mehr steif. Der globale Fehler der so erhaltenen numerischen Approximation enthält einenɛ-abhängigen Term, der mitɛ gegen 0 geht. Das bedeutet, dass der Fehler nicht zunimmt mit der Steifheit des Systems. Des weiteren wird die Implementation der Methode diskutiert und eine Anwendung gegeben auf ein nichttriviales Beispiel.

CSE program at ETH Zurich: are we doing the right thing?

Computational techniques, next to theory and experiments, have become a major and widespread tool to solve large problems in Science and Engineering. Especially in places where large computer power was available, such as in North America, Japan and Europe, the need was recognized to start to educate persons in the new discipline Computational Science and Engineering (CSE). A first survey of the state of affairs in this new subject is given in the report ’Graduate Education in Computational Science and Engineering’ of the SIAM working Group [3].

Numerical integration of differential algebraic systems and invariant manifolds

In 1997 a new interdisciplinary Diploma program in Computational Science and Engineering (CSE) was started at ETH Zurich. We report on the changes of the curriculum due to the Bologna Declaration of June 19, 1999 by the European Ministers of Education. The new Bachelor program in CSE at ETH Zurich started in October 2003 and the Master program in CSE will follow in October 2005. We will describe both programs in some detail as well as the design principles. We will also discuss the boundary conditions imposed by ETH and we will give some preliminary experiences.