Regularization of nonlinear DAEs based on Structural Analysis

Differential algebraic equations (henceforth DAEs), arise from the equation based modelling of physical systems, such as those found in engineering or physics applications, with problems specifically arising from chemical distillation (Washington and Swartz (2011)), electronic circuits (Brenan et al. (1996)) and robotics (Campbell and Griepentrog (1995)). Models are now frequently built interactively using different components from large libraries in environments such as GPROMS, MAPLESIM, SIMULINK and an assortment of tools that use the Modelica language. This way of modelling systems can lead to large scale DAEs (Fritzson (2015)). A common notion is the differentiation index—which is equal to the number of times all or part of the system has to be differentiated in order to solve the problem as an ODE. It is well known that solving a high index (larger than one) DAE directly is numerically difficult, hence modelling tools usually perform some structural analysis to determine the index of the problem. This talk will outline different ways of using this structural analysis in the solution and regularization of DAEs. In particular the Signature Matrix method (Pryce (2001)), Dummy Derivative method (Mattsson and Söderlind (1993)), Structural-Algebraic method (Scholz and Steinbrecher (2013)) and a new approach called Structural Analysis based Dummy Derivatives will be considered.