# An asymptotic preserving mixed finite element method for wave propagation in pipelines

## TL;DR

This paper develops an asymptotic preserving mixed finite element method for wave propagation in pipelines, effectively capturing the transition to steady states and parabolic limits with proven stability and accuracy.

## Contribution

It introduces a novel mixed finite element approach that preserves asymptotic behavior in the parabolic limit and demonstrates uniform exponential stability regardless of discretization.

## Key findings

- Method is asymptotic preserving and uniformly exponentially stable.
- Numerical tests confirm sharpness of theoretical estimates.
- Approach effectively handles parameter-dependent damped hyperbolic equations.

## Abstract

We consider a parameter dependent family of damped hyperbolic equations with interesting limit behavior: the system approaches steady states exponentially fast and for parameter to zero the solutions converge to that of a parabolic limit problem. We establish sharp estimates and elaborate their dependence on the model parameters. For the numerical approximation we then consider a mixed finite element method in space together with a Runge-Kutta method in time. Due to the variational and dissipative nature of this approximation, the limit behavior of the infinite dimensional level is inherited almost automatically by the discrete problems. The resulting numerical method thus is asymptotic preserving in the parabolic limit and uniformly exponentially stable. These results are further shown to be independent of the discretization parameters. Numerical tests are presented for a simple model problem which illustrate that the derived estimates are sharp in general.