Timestepping Schemes Based On Time Discontinuous Galerkin Methods

The contribution deals with timestepping schemes for nonsmooth dynamical systems. Traditionally, these schemes are locally of integration order one, both in smooth and nonsmooth periods. This is inefficient for applications with few events like circuit breakers, valve trains or slider-crank mechanisms. To improve the behavior during smooth episodes, we start activities twofold. First, we include the classic schemes in time discontinuous Galerkin methods. Second, we split smooth and nonsmooth force propagation. The correct mathematical setting is established with mollifier functions, Clenshaw-Curtis quadrature rules and appropriate impact representation. The result is a Petrov-Galerkin distributional differential inclusion. It defines two Runge-Kutta collocation families and enables higher integration order during smooth transition phases. As the framework contains the classic Moreau-Jean timestepping schemes for constant ansatz and test functions on velocity level, it can be considered as a consistent enhancement. An experimental convergence analysis with the bouncing ball example illustrates the capabilities.