# Multirate numerical integration of ordinary differential equations

Numerous phenomena from different areas of science and technology are modelled by systems of ordinary differential equations (ODEs). ODEs describe the motion of a body by its position and velocity; the evolution of the current in an electrical circuit; the change of the temperature of an object in a given environment; and even the dynamics of the price of a stock.

Publication date: 16-01-2008

Numerous phenomena from different areas of science and technology are modelled by systems of ordinary differential equations (ODEs). ODEs describe the motion of a body by its position and velocity; the evolution of the current in an electrical circuit; the change of the temperature of an object in a given environment; and even the dynamics of the price of a stock.

For the numerical solution of systems of ODEs there are many methods available. These methods use time steps that are varying in time, but are constant over the components. However, there are many problems of practical interest, where the temporal variations have different time scales for different sets of the components. To exploit these local time scale variations, one needs multirate methods that use different, local time steps over the components. In these methods big time steps are used for the slow components and small time steps are used for the fast ones.

In his PhD dissertation "Multirate numerical integration of ordinary differential equations", Valeriu Savcenco (CWI) designed, analyzed and tested multirate methods for the numerical solution of ODEs. He developed a self-adjusting multirate time stepping strategy, in which the step size for a particular system component is determined by the local temporal variation of this solution component, in contrast to the use of a single step size for the whole set of components as in the traditional methods. The partitioning into different levels of slow to fast components is performed automatically during the time integration.

Savcenco received his PhD on 15 January 2008 at the University of Amsterdam. The research has been carried out at CWI and was financed with a Peter Paul Peterich scholarship from NWO. This scholarship is provided once every four years for the best math research proposal. More information about this research can be found on the homepage of Valeriu Savcenco. NWO posted a summary in Dutch on its website.