Simulation and Optimization of Partial Differential-Algebraic Equations with a Separation of Time Scales

Problems that exhibit multiple time scales arise frequently in many scientific and engineering fields. The modeling and simulation of such systems leads naturally to singular perturbation models. For systems of partial differential-algebraic equations (PDAEs) in time and one spatial dimension, the corresponding quasi-steady-state models yields a reduced set of PDAEs (slow variables), subject to a set of differential-algebraic equations (DAEs) in the spatial dimension (fast variables). In this presentation, we shall consider a particular class of one-dimensional, quasi-linear PDAEs with a separation of time scales such that (i) the slow variables are lumped (i.e., do not depend on the spatial dimension), and (ii) the hyperbolic variables in the fast subsystem have all their characteristics pointing in the same direction. Under these conditions, the quasi-steady-state model yields two decoupled subsystems, a set of DAEs in time, subject to a set of DAEs in the spatial dimension; hence the name `DAEs embedded DAEs'. There are several advantages in using the DAEs embedded DAEs approach over the conventional method of lines (MOL) for such problems. First and foremost, this approach guarantees the accuracy of the solution, in the limit of the slow model approximation validity, since rigorous error control can b e performed by numerical solvers regarding the time and space steps used in either set of differential equations. This removes the need of choosing a somewhat arbitrary discretization as it is the case with the MOL. Furthermore, the DAEs embedded DAEs approach requires solution of much smaller sets of differential equations than with the MOL approach. Therefore, not only does the proposed appr oach improve the reliability of the simulations by removing the need of initializing large sets of DAEs, but it also typically outperforms the conventional MOL in terms of computational time whenever the use of fine meshes becomes necessary. This also makes the DAEs embedded DAEs simulation approach particularly well suited for embedding within a mathematical programming formulation for optimi zation purposes. The developed approach shall be demonstrated on an application related to the start-up simulation a nd optimizaton of micro-scale chemical processes for portable power generation.