On the use of reduced order models in bifurcation analysis of distributed parameter systems

Bifurcation theory provides a powerful tool for analyzing the nonlinear dynamic behavior of process systems. However, although the theory in principle applies to lumped as well as distributed parameter processes, it is in practice necessary to reduce the order of distributed (partial differential equations, PDE) models prior to application of the theory. As shown in this paper, simply applying some ad hoc discretization method such as finite differences or finite elements, can result in spurious bifurcations and erroneous predictions of stability. To enable detection of such anomalities, and to aid in the selection of a proper model order, we propose a method for estimating the error introduced by the model reduction. Apart from simply providing a label of confidence in the results of the bifurcation analysis, the estimated error can be used to improve the quality of the reduced order model. For this purpose we propose a method based on dynamically moving the discretization mesh such as to minimize the discretization error. The proposed method is based on principles from feedback control, and is both very simple and highly robust compared with existing so-called moving mesh methods. As an application we consider bifurcation analysis of a heat-integrated fixed-bed reactor.