Efficient integration over discontinuities for differential-algebraic systems

In recent years, hybrid (discrete/continuous) dynamic systems that exhibit coupled continuous and discrete behavior have attracted much attention. Many engineering problems can be formulated as ODE/DAEs which can be solved by numerical methods. In process design, dynamic optimization and simulation, however, many systems of interest experience significant discontinuities during transients. This paper describes some simple strategies for discontinuity detecting and handling in DAE embedded systems. The described algorithm supports flexible representation of state conditions in propositional logic. By making full use of the discontinuity functions, both efficiency of integration and effectiveness of discontinuity detection are achieved. Considerable care is taken to handle problems that can arise during mode transitions. We outline the implementation in a general-purpose solver DASPKE for differential-algebraic equations containing discontinuities, and present some numerical experiments which illustrate its effectiveness.