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

Tommaso Tenna

TL;DR

This work develops a relaxation BGK-type numerical scheme for hyperbolic conservation laws on canal and arterial networks, enabling junction solutions without solving local Riemann problems. The method couples a discrete kinetic relaxation with well-balanced treatments for source terms, addressing both subcritical and supercritical regimes in shallow water and arterial blood flow models. Entropy dissipation, mass conservation, and positivity are established, and the approach is validated through extensive network simulations (including bottlenecks and complex junctions) that align with traditional Riemann-solver results. The framework provides accurate, efficient junction handling with robust steady-state preservation, suggesting broad applicability to networked hyperbolic systems and potential extensions to other transport phenomena.

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.

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

TL;DR

This work develops a relaxation BGK-type numerical scheme for hyperbolic conservation laws on canal and arterial networks, enabling junction solutions without solving local Riemann problems. The method couples a discrete kinetic relaxation with well-balanced treatments for source terms, addressing both subcritical and supercritical regimes in shallow water and arterial blood flow models. Entropy dissipation, mass conservation, and positivity are established, and the approach is validated through extensive network simulations (including bottlenecks and complex junctions) that align with traditional Riemann-solver results. The framework provides accurate, efficient junction handling with robust steady-state preservation, suggesting broad applicability to networked hyperbolic systems and potential extensions to other transport phenomena.

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.

Paper Structure

This paper contains 35 sections, 3 theorems, 137 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Models
  3. The Shallow Water Equations
  4. The Arterial Network
  5. Flows in Canal Networks
  6. Entropy Dissipation
  7. Shallow Water Equations
  8. Blood Flow Equations
  9. Relaxation schemes on Networks
  10. Discrete kinetic schemes
  11. Discrete kinetic schemes on a 2-canal network
  12. Numerical treatment of the junction
  13. Well-balanced schemes
  14. Shallow Water Equations
  15. Blood Flow Equations
  16. ...and 20 more sections

Key Result

Proposition 4.2

Let us consider the numerical scheme Relaxation_Final on a 2-canal network, with junction condition given by junction_mass_cons_shallow-junction_pressure_shallow and boundary conditions boundary_ingoing_discrete-boundary_outgoing_discrete. The scheme is mass-preserving, namely Moreover, if the kinetic velocities are chosen such that and the initial height (or section $a_i$) is positive, then the

Figures (16)

  • Figure 1: Simple 2-canal network with one incoming and one outgoing branch, where $J$ denotes the junction.
  • Figure 2: (Sanity Check) Riemann problem \ref{['Initial_Condition_Test1']} in $x \in (-4,4)$ with a virtual junction in $x=0$.
  • Figure 3: (Bottleneck) Riemann problem \ref{['Initial_Condition_Test1']} with two different cross sections in $x \in (-2,4)$ with a junction in $x=1$.
  • Figure 4: (Bottleneck) Riemann problem \ref{['Initial_Condition_Test3']} in $x \in (-2,4)$ with a junction in $x=1$.
  • Figure 5: (Bottleneck) Riemann problem \ref{['SW_Bottleneck_FT']} in $x \in (-2,2)$ with a junction in $x=0$ for a supercritical case.
  • ...and 11 more figures

Theorems & Definitions (7)

  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • proof