Multirate Schemes — An Answer of Numerical Analysis to a Demand from Applications

Abstract

AbstractIn science and engineering, simulation tasks often involve numerical time integration of differential equations. Usually, these systems contain different time constants of the involved components and/or right-hand side. This multirate behavior may be caused by coupling subsystems in multiphysics problems acting on different time scales. Such a behavior does already occur if one deals with just single-physics problems: for example, the activity level of components in electrical networks may strongly vary depending on the according functional purpose, physics or time; another example is given in lattice QCD, where the equations of motion may depend on weak and strong forces, which demand to sample these forces with different frequencies to gain the same rate of approximation.To be efficient or to speed up simulation of highly complex coupled systems is necessary for many design and optimization work flows. To this end, numerical integration schemes have to be adapted to exploit this multirate behavior. One idea proposed by Rice in 1960 are multirate schemes, which use different step sizes adapted to the various activity levels. In the last 50 years, the methodology of numerical time integration schemes has been advanced in a constant interplay between the demands defined by the need of exploiting multirate behavior in different fields of applications and the development of tailored multirate schemes to answer these demands.