Relaxation Schemes for Flows in Networks: Application to Shallow Water and Blood Flow Equations

TL;DR

This paper introduces a relaxation-based numerical scheme for hyperbolic conservation laws in networks, effectively handling shallow water and blood flow equations without Riemann solvers, and includes a well-balanced approach for source terms.

Contribution

It presents a novel relaxation scheme for network flows that avoids Riemann solvers and applies to both shallow water and blood flow equations, including source term treatment.

Findings

Scheme effectively handles subcritical and supercritical flows.

Numerical tests demonstrate accuracy and robustness.

Well-balanced strategy improves source term approximation.

Abstract

A numerical scheme of relaxation type is proposed to approximate hyperbolic conservation laws in canal networks. Physical conditions at the junction are given and a novel strategy based on [Briani, Natalini, Ribot, 2025] is introduced to approximate the solution, avoiding the use of approximate Riemann solvers. This general approach is applied to shallow water and blood flow equations, dealing both the subcritical and the supercritical case. The relaxation scheme is complemented with a well-balanced strategy to treat source terms. We investigate properties of the numerical scheme and we present many numerical tests in different settings.