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arxiv: 2509.22190 · v1 · submitted 2025-09-26 · 🧮 math.NA · cs.NA

Well-balanced high-order method for non-conservative hyperbolic PDEs with source terms: application to one-dimensional blood flow equations with gravity

Chiara Colombo , Caterina Dalmaso , Lucas O. M\"uller , Annunziato Siviglia This is my paper

Pith reviewed 2026-05-18 12:39 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords well-balanced methodsnon-conservative hyperbolic PDEsfinite volume schemesblood flow modelingsource termshigh-order accuracygravity and friction
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The pith

A high-order well-balanced method preserves stationary solutions of blood flow equations with gravity to machine precision.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a finite volume method for non-conservative hyperbolic PDEs that include source terms such as gravity and friction. It combines a compact high-order spatial reconstruction, drawn from generalized Riemann problem data at the previous time level, with a well-balanced space-time evolution operator to support fully explicit time stepping. Tests confirm that the scheme holds numerically computed stationary states exactly while achieving the expected convergence rates, first on Burgers' equation and then on a hyperbolized one-dimensional blood flow model with variable mechanical and geometrical properties. The approach is also demonstrated on a network of 86 arteries under both steady and time-varying conditions. This matters for long simulations where small imbalances between fluxes and sources would otherwise accumulate into large errors.

Core claim

The central claim is that the combination of a recently introduced high-order reconstruction, well-balanced up to order three, and a well-balanced space-time evolution operator yields a method that preserves stationary solutions of the target system up to machine precision and attains the designed order of accuracy in both space and time on the hyperbolized blood flow equations with gravity, friction, and spatially varying coefficients.

What carries the argument

High-order spatial reconstruction based on generalized Riemann problem information from the previous time level, together with a well-balanced space-time evolution operator.

If this is right

  • Stationary solutions on arterial networks remain accurate without cumulative numerical drift.
  • The scheme retains high-order convergence even when vessel geometry and elasticity change along the domain.
  • Fully explicit time evolution stays well-balanced for both steady and transient regimes.
  • The same construction applies directly to other non-conservative hyperbolic systems with geometric or mechanical source terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could reduce the need for artificial damping or implicit corrections in long-term physiological flow simulations.
  • Extending the reconstruction to order four or higher would require checking whether the well-balanced property continues to hold under spatial variation of coefficients.
  • Similar well-balanced reconstructions may be useful for related models that couple blood flow with vessel wall mechanics or external fields.

Load-bearing premise

The high-order reconstruction using information from the prior time level remains well-balanced when the system coefficients vary in space.

What would settle it

Integrate the scheme on a mesh with known analytical stationary solution for the blood flow equations and check whether the discrete solution drifts from exact balance by more than round-off error after many steps.

Figures

Figures reproduced from arXiv: 2509.22190 by Annunziato Siviglia, Caterina Dalmaso, Chiara Colombo, Lucas O. M\"uller.

Figure 1
Figure 1. Figure 1: Burgers’ problem. Initial condition (A), steady-state solution (B), and errors in space between the numerical solution and the steady-state solu￾tion obtained either numerically (C and D) or analytically (E and F). Results are shown for both second- and third-order implementations of the GRP+DET and the GRP+DET-WB methods. condition (30) is given by a small spatial perturbation of that stationary solu￾tion… view at source ↗
Figure 2
Figure 2. Figure 2: Efficiency plots for ICA test. CPU times versus L 2 error norms between the numerical solution and either the exact solution (S1 and S2) or a reference solution (S3), for all the considered scenarios and for 4 consecutive mesh refinements. Each row refers to a different scenario (S1, S2, and S3). Results are shown for both cross-sectional area (left panels) and flow rate (right panels) in logarithmic scale… view at source ↗
Figure 3
Figure 3. Figure 3: Deadman test. A representation of the ADAN86 network is shown. In the top panel, the colors indicate the errors between zero-flow solution and the second-order numerical solution computed with either the GRP+DET-WB, or the GRP+DET. In the bottom panel, the colors indicate the errors between the reference hydrostatic pressure distribution and the numerical solution computed with either the GRP+DET-WB, or th… view at source ↗
Figure 4
Figure 4. Figure 4: Pressure distribution. Pressure distribution at the final simulation time along the ADAN86 network obtained with a second-order implementation of both the GRP+DET-WB method (left) and the GRP+DET method (right). A focus on eight vessels is provided in the middle of the panel, showing how the GRP+DET-WB results respect the expected hydrostatic distribution indicated in light gray. The considered vessels are… view at source ↗
Figure 5
Figure 5. Figure 5: Transient test. Flow rate (left column) and pressure (right column) curves along the final cardiac cycle of simulation at the midpoint of three selected blood vessels: left anterior cerebral artery (orange), thoracic aorta (red), and left femoral artery (blue). Indication on the location of the vessels is shown in the ADAN86 network representation on the left. Black dashed lines represent the reference sol… view at source ↗
Figure 6
Figure 6. Figure 6: Wave configuration. Representation of the wave configuration in the x − t half plane. Our goal is to solve this RP under the condition that the flow is sub￾critical. Assuming a subcritical flow is equivalent to asking that the fluid velocity u is smaller than the wave propagation speed c, namely λ1 < 0 and λ8 > 0. Indeed, we recall that the hyperbolized BFEs present 8 waves [39], with associated eigenvalue… view at source ↗
Figure 7
Figure 7. Figure 7: Deadman test. A representation of the ADAN86 network is shown. In the top panel, the colors indicate the errors between zero-flow solution and the third-order numerical solution computed with either the GRP+DET-WB, or the GRP+DET. In the bottom panel, the colors indicate the errors between the reference hydrostatic pressure distribution and the numerical solution computed with either the GRP+DET-WB, or the… view at source ↗
Figure 8
Figure 8. Figure 8: Pressure distribution. Pressure distribution at the final simulation time along the ADAN86 network obtained with a third-order implementation of both the GRP+DET-WB method (left) and the GRP+DET method (right). A focus on eight vessels is provided in the middle of the panel, showing how the GRP+DET-WB results respect the expected hydrostatic distribution indicated in light gray. The considered vessels are:… view at source ↗
Figure 9
Figure 9. Figure 9: Transient test. Flow rate (left column) and pressure (right column) curves along the final cardiac cycle of simulation at the midpoint of three selected blood vessels: left anterior cerebral artery (orange), thoracic aorta (red), and left femoral artery (blue). Indication on the location of the vessels is shown in the ADAN86 network representation on the left. Black dashed lines represent the reference sol… view at source ↗
read the original abstract

The present work proposes a well-balanced finite volume-type numerical method for the solution of non-conservative hyperbolic partial differential equations (PDEs) with source terms. The method is characterized, first, by the use of a recently introduced high-order spatial reconstruction, based on generalized Riemann problem information from the previous time level. Such reconstruction is well-balanced up to order three, compact, efficient and easy to implement. Second, the method incorporates a well-balanced space-time evolution operator, which allows for well-balanced fully explicit time evolution. The accuracy and efficiency of the method are assessed on both a scalar problem (Burgers' equation) and a nonlinear PDE system (hyperbolized one-dimensional blood flow equations with gravity and friction, and with variable mechanical and geometrical properties). The well-balanced property is verified by showing that numerically-determined stationary solutions are preserved up to machine precision. The order of accuracy in space and time is validated through empirical convergence rate studies. Additionally, the performance of the method is assessed on a network of 86 arteries, under both stationary and transient conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper proposes a well-balanced finite-volume method for non-conservative hyperbolic PDEs with source terms. It combines a recently introduced high-order spatial reconstruction (based on generalized Riemann problem data from the prior time level, claimed well-balanced up to order three) with a well-balanced space-time evolution operator that permits fully explicit time stepping. The scheme is tested on Burgers' equation and on the hyperbolized one-dimensional blood-flow system that includes gravity, friction, and spatially varying mechanical and geometrical properties. Stationary solutions are shown to be preserved to machine precision and empirical convergence rates matching the design order are reported; the method is also demonstrated on an 86-artery network under both steady and transient conditions.

Significance. If the central well-balanced property holds for variable-coefficient non-conservative systems, the approach would offer a practical, high-order, fully explicit alternative for balance laws arising in hemodynamics and related fields. The numerical evidence of machine-precision preservation of discrete equilibria and observed convergence rates on the target blood-flow equations constitutes a concrete strength; the explicit treatment and compact reconstruction are also attractive for large-scale network simulations.

major comments (1)
  1. [§4.2] §4.2 (blood-flow discretization): the well-balanced property of the order-three reconstruction for the non-conservative product and variable-coefficient source terms is asserted on the basis of the reconstruction's design and verified numerically, but the manuscript does not supply an explicit algebraic verification that the reconstruction operator exactly reproduces the steady-state relation when the geometry and mechanical properties vary spatially; a short derivation or counter-example check would remove any residual doubt about the load-bearing assumption.
minor comments (3)
  1. [§3.1] The description of the generalized Riemann problem reconstruction in §3.1 would benefit from an explicit statement of the stencil width and the precise polynomial degree used at each interface.
  2. [Figure 7] Figure 7 (network results) lacks error bars or a quantitative comparison against a reference solution; adding at least one such metric would clarify the practical accuracy on the 86-artery test.
  3. [§2.3] A few typographical inconsistencies appear in the notation for the friction term between the abstract and §2.3; harmonizing the symbols would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comment. We address the major comment below.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (blood-flow discretization): the well-balanced property of the order-three reconstruction for the non-conservative product and variable-coefficient source terms is asserted on the basis of the reconstruction's design and verified numerically, but the manuscript does not supply an explicit algebraic verification that the reconstruction operator exactly reproduces the steady-state relation when the geometry and mechanical properties vary spatially; a short derivation or counter-example check would remove any residual doubt about the load-bearing assumption.

    Authors: We thank the referee for highlighting this point. The well-balanced character of the order-three reconstruction follows directly from its construction, which embeds the steady-state relations of the target system into the generalized Riemann problem data from the prior time level. Nevertheless, we agree that an explicit algebraic verification for the case of spatially varying cross-sectional area, wall stiffness, and gravity would strengthen the exposition and eliminate any residual ambiguity. In the revised manuscript we will insert a short derivation in §4.2 that substitutes the discrete equilibrium conditions into the reconstruction formulas and verifies exact cancellation for the non-conservative product and source terms. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of reconstruction; central claims verified independently on target system

full rationale

The derivation begins from the stated properties of a recently introduced high-order spatial reconstruction (well-balanced up to order three) combined with a well-balanced space-time evolution operator to form the finite-volume scheme. The central well-balanced property for the non-conservative blood-flow system with variable coefficients, gravity, and friction is then confirmed by direct numerical experiments: machine-precision preservation of discrete stationary solutions and empirical convergence rates matching the design order. These tests operate on the full target equations without relying on fitted constants internal to the scheme or on any self-citation chain that would render the result tautological. The single likely self-citation of the reconstruction technique is therefore not load-bearing for the paper's primary claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the well-balanced property of the imported reconstruction and on the correctness of the space-time evolution operator; both are treated as established inputs rather than re-derived here.

axioms (1)
  • domain assumption The high-order spatial reconstruction based on generalized Riemann problem information is well-balanced up to order three for the target non-conservative system.
    Invoked when the method is characterized in the abstract as using this reconstruction.

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Reference graph

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