Time discretization schemes for hyperbolic systems on networks by $\hat{I}\mu$-expansion

Abstract

We consider partial differential equations on networks with a small parameter $\hat{I}\mu$, which are hyperbolic for $\hat{I}\mu >0$ and parabolic for $\hat{I}\mu =0$. With a combination of an $\hat{I}\mu$-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 $\hat{I}\mu$-expansion.