# On structure-preserving model reduction for damped wave propagation in transport networks

## TL;DR

This paper develops structure-preserving model reduction techniques for damped wave equations in pipeline networks, ensuring key physical properties are maintained during discretization and reduction, with practical algorithms based on Krylov subspaces.

## Contribution

It introduces a novel structure-preserving Galerkin projection combined with Krylov subspace methods for model reduction of PDE-DAE systems in pipeline networks.

## Key findings

- Reduced models preserve energy dissipation and passivity.
- Algorithms ensure well-posedness and stability of reduced systems.
- Numerical tests confirm the effectiveness of the proposed methods.

## Abstract

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.