On structure-preserving model reduction for damped wave propagation in transport networks
For engineers modeling transport networks, this work provides a method to reduce model complexity while preserving essential physical properties, though it is incremental as it extends existing Galerkin projection and Krylov subspace techniques.
This paper develops structure-preserving model reduction for damped wave propagation in pipeline networks, ensuring conservation of mass, energy dissipation, passivity, and stability. Numerical tests demonstrate the necessity of compatibility conditions for well-posedness.
We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semi-discretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construciton of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic equations while the intermediate discretization by finite elements is only required for the actual computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples.