On the role of quadrature rules and system dimensions in variational multirate integrators

Systems like a chain driven combustion engine or a satellite in space are mechanical systems containing slow and fast dynamics. Multirate schemes split such systems into parts which can be simulated using different time steps. With the framework of variational multirate integration a time grid with micro and macro nodes is introduced. Simulating the slow variables on the macro grid and the fast variables on the micro grid leads to computing time savings. However, increasing the number of micro steps per macro step leads to a larger system of equations that has to be solved iteratively. Therefor, there is a limit to the computing time savings. The relation between system size, numbers of micro steps and computational effort is investigated by means of the Fermi-Pasta-Ulam problem. Therefor, the computing time for different numbers of masses for the Fermi-Pasta-Ulam problem is investigated for an increasing number of micro nodes and for different quadrature schemes. Relations between the minimum computing time and the numbers of variables are derived. Furthermore, another aspect is studied. The convergence rate for different quadrature schemes is investigated numerically for the Fermi-Pasta-Ulam problem as an unconstrained system and the triple spherical pendulum as a system subject to constraints.