An asymptotic preserving mixed finite element method for wave propagation in pipelines
This work provides a robust numerical method for wave propagation in pipelines, ensuring stability across parameter regimes, but the results are incremental as they extend existing asymptotic preserving techniques to a specific application.
The paper develops an asymptotic preserving mixed finite element method for damped hyperbolic equations that converge to a parabolic limit, achieving uniform exponential stability and sharp parameter-dependent estimates. Numerical tests confirm the sharpness of the estimates.
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.