Time discretization schemes for hyperbolic systems on networks by $\epsilon$-expansion

TL;DR

This paper develops new time discretization schemes for hyperbolic PDEs on networks with small parameters, using epsilon-expansion and Runge-Kutta methods, ensuring well-posedness and error estimation.

Contribution

It introduces a novel class of schemes combining epsilon-expansion with Runge-Kutta methods for constrained hyperbolic systems on networks.

Findings

New discretization schemes for hyperbolic systems on networks.

Error estimates for epsilon-expansion approximations.

Analysis of well-posedness of the coupled PDE-DAE systems.

Abstract

We consider partial differential equations on networks with a small parameter $\epsilon$, which are hyperbolic for $\epsilon>0$ and parabolic for $\epsilon=0$. With a combination of an $\epsilon$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $\epsilon$-expansion.