Chementator: New methods for solving differential equations  

The behavior of many different phenomena can be modeled by systems of ordinary differential equations (ODEs), but most ODEs cannot be solved analytically, in which case an approximation to the solution is found by applying numerical integration methods. Dutch-sponsored mathematician Valeriu Savcenco has developed new "multirate" numerical methods that can significantly reduce the CPU time and effort required to reach a solution compared to commonly used methods, such as trapezoidal, Euler and Runge-Kutta. Common numerical 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 components, says Savcenco. For example, cell phones consist of coupled digital and analog sub-circuits, which operate in nano- and microseconds. To exploit these local time-scale variations, one needs multirate methods that use different, local time steps over the components; big time steps are used for the slow components, and small time steps are used for the fast ones. With the multirate method, improvement factors of 6, 8 and 10 have been achieved for the problems Savcenco solved for…