High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models

TL;DR

This paper develops high order finite volume schemes for networks of hyperbolic conservation laws coupled with ODEs, ensuring conservation and accurate shock capturing, and introduces a technique for integrating lumped parameter models.

Contribution

It introduces novel high order numerical methods for coupled hyperbolic PDEs and ODEs on networks, including techniques for lumped parameter models.

Findings

High order convergence demonstrated in numerical tests.

Exact conservation achieved with the proposed coupling schemes.

Robust shock capturing in complex network scenarios.

Abstract

In this paper we construct high order finite volume schemes on networks of hyperbolic conservation laws with coupling conditions involving ODEs. We consider two generalized Riemann solvers at the junction, one of Toro-Castro type and a solver of Harten, Enquist, Osher, Chakravarthy type. The ODE is treated with a Taylor method or an explicit Runge-Kutta scheme, respectively. Both resulting high order methods conserve quantities exactly if the conservation is part of the coupling conditions. Furthermore we present a technique to incorporate lumped parameter models, which arise from simplifying parts of a network. The high order convergence and the robust capturing of shocks is investigated numerically in several test cases.