Time discretization schemes for hyperbolic systems on networks by $ε$-expansion
This work provides a novel numerical method for solving hyperbolic PDEs on networks, which is relevant for applications in network flow problems, but the results are theoretical and no concrete performance numbers are given.
The paper develops new time discretization schemes for hyperbolic systems on networks with a small parameter ε by combining ε-expansion with Runge-Kutta methods for constrained parabolic systems. The approach addresses the challenge of interconnection conditions and provides error estimates for the ε-expansion.
We consider partial differential equations on networks with a small parameter $ε$, which are hyperbolic for $ε>0$ and parabolic for $ε=0$. With a combination of an $ε$-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 $ε$-expansion.