Generalized Hydrostatic Reconstruction

• Generalized hydrostatic reconstruction is a framework that adapts interface states to cancel steady-state residuals in hyperbolic balance laws.
• It unifies approaches in blood flow and shallow water modeling by reconstructing interface values and correcting source discretization to preserve target equilibria.
• Different variants, such as HR–LS and HR–S, offer trade-offs between computational efficiency and exact well-balanced accuracy for subcritical regimes.

Searching arXiv for the cited hydrostatic reconstruction papers to ground the article in published work. Generalized hydrostatic reconstruction denotes a family of well-balanced reconstruction procedures for hyperbolic balance laws with geometric, topographic, or material heterogeneity, in which interface states are reconstructed so that a prescribed steady system is preserved exactly, or asymptotically in a targeted regime, rather than only the classical rest equilibrium. In the formulation for one-dimensional blood flow in elastic arteries with varying geometrical and mechanical properties, the method replaces the classical hydrostatic reconstruction around the “dead-man” equilibrium by reconstructions adapted to low-Shapiro or fully subcritical steady states (Ghigo et al., 2017). In shallow-water settings, related constructions preserve lake-at-rest states, moving steady states with nonzero discharge, or entropy-stable path-conservative equilibria on wet/dry and multilayer configurations (Berthon et al., 7 Jan 2025, Ersing et al., 2024). This suggests a unifying interpretation of generalized hydrostatic reconstruction as a reconstruction-plus-source-discretization framework designed to cancel the discrete residual of a target steady manifold while retaining conservative flux differencing.

1. Definition and scope

The classical hydrostatic reconstruction (HR) is a first-order well-balanced procedure originally organized around hydrostatic rest states. For shallow water with bottom $z(x)$, the Saint–Venant system is written as

$$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$

and the HR strategy reconstructs interface states $(h^\pm_{i+1/2},Q^\pm_{i+1/2})$ so as to exactly balance the source $g h\,z_x$ (Kivva, 2023). In one-dimensional blood flow, the corresponding balance law includes variable rest area $A_0(x)$ and wall rigidity $K(x)$, with a nonconservative source produced by spatial variations of these quantities (Ghigo et al., 2017).

A plausible unifying implication is that “generalized” HR is not a single algorithm but a design principle: select reconstructed interface states and source corrections so that the discrete scheme annihilates the steady-state residual relevant to the application. In blood flow, the target may be a low-Shapiro or fully subcritical steady system (Ghigo et al., 2017). In shallow water, the target may be lake-at-rest, moving Bernoulli equilibria with nonzero velocity, or entropy-compatible equilibria in one-layer and multilayer systems (Berthon et al., 7 Jan 2025, Ersing et al., 2024).

The same family resemblance appears in the numerical update. In the blood-flow framework, one reconstructs

$$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$

then defines a well-balanced flux

$$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$

with well-balancing ensured by the choice of $\mathcal R^*$ and consistent source terms (Ghigo et al., 2017). The same pattern recurs in shallow-water formulations that combine reconstructed interface states with suitable numerical fluxes and discrete source terms (Berthon et al., 7 Jan 2025, Ersing et al., 2024).

2. Blood-flow formulation and low-Shapiro reconstruction

For inviscid one-dimensional blood flow in an elastic artery, the conservative-plus-source form is

$$\frac{\partial A}{\partial t} +\frac{\partial Q}{\partial x} =0, \qquad \frac{\partial Q}{\partial t} +\frac{\partial}{\partial x}\Bigl(\frac{Q^2}{A}+\frac{K}{3\rho}\,A^{3/2}\Bigr) =\frac{A}{\rho}\Bigl[\partial_x(K\sqrt{A_0})-\tfrac{2}{3}\sqrt{A}\,\partial_xK\Bigr].$$

Here $$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$0 is the cross-sectional area, $$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$1 the volume flow rate, $$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$2 the mean velocity, $$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$3 the blood density, and the wall law is $$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$4 (Ghigo et al., 2017). The momentum flux is

$$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$5

the Moens–Korteweg wave speed is

$$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$6

and the Shapiro number is

$$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$7

Physiological flows are strongly subcritical, $$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$8, and the steady states at rest are stated not to be relevant for blood flow (Ghigo et al., 2017).

This motivates the low-Shapiro hydrostatic reconstruction (HR–LS). The exact subcritical steady state satisfies

$$\partial_t h + \partial_x Q = 0,\qquad \partial_t Q + \partial_x\Bigl(\tfrac{Q^2}{h} + \tfrac{g\,h^2}{2}\Bigr) = -\,g\,h\,\partial_x z,$$9

In the low-$(h^\pm_{i+1/2},Q^\pm_{i+1/2})$0 limit, the quadratic advection term is dropped, yielding

$(h^\pm_{i+1/2},Q^\pm_{i+1/2})$1

At each interface $(h^\pm_{i+1/2},Q^\pm_{i+1/2})$2, with left/right states $(h^\pm_{i+1/2},Q^\pm_{i+1/2})$3 and $(h^\pm_{i+1/2},Q^\pm_{i+1/2})$4, HR–LS reconstructs $(h^\pm_{i+1/2},Q^\pm_{i+1/2})$5 and $(h^\pm_{i+1/2},Q^\pm_{i+1/2})$6 so that

$(h^\pm_{i+1/2},Q^\pm_{i+1/2})$7

with positivity $(h^\pm_{i+1/2},Q^\pm_{i+1/2})$8. A convenient choice is

$(h^\pm_{i+1/2},Q^\pm_{i+1/2})$9

followed by

$g h\,z_x$0

and analogously on the right (Ghigo et al., 2017).

The resulting numerical flux is

$g h\,z_x$1

where $g h\,z_x$2 and

$g h\,z_x$3

This construction exactly preserves the low-Shapiro steady system (Ghigo et al., 2017).

3. Exact subcritical preservation and the generalized operator

The subsonic hydrostatic reconstruction (HR–S) extends the same idea to the full subcritical steady system. In that case the target steady relation is

$g h\,z_x$4

Introducing

$g h\,z_x$5

the interface reconstruction imposes

$g h\,z_x$6

with analogous equations on the right and again

$g h\,z_x$7

Because $g h\,z_x$8 is convex in $g h\,z_x$9, one numerically solves for the unique subcritical root $A_0(x)$0, where $A_0(x)$1 minimizes $A_0(x)$2 (Ghigo et al., 2017).

The abstract operator viewpoint is explicit in the blood-flow formulation. In both HR–LS and HR–S, the key difference from classical HR is that reconstruction is performed around the appropriate subcritical steady system rather than around $A_0(x)$3 and $A_0(x)$4. This is the sense in which the framework is “generalized”: the reconstruction $A_0(x)$5 is chosen so that the discrete residual of the target steady system vanishes, and the well-balanced flux $A_0(x)$6 adds source corrections that exactly cancel the truncation error between flux differences and source discretization at that target steady state (Ghigo et al., 2017).

A concise comparison of the three blood-flow reconstructions is given below.

Method	Preserved steady state	Reported characteristics
HR	Rest-state reconstruction	Not adapted to compute blood flow in large arteries when large variations of geometrical and mechanical properties are considered
HR–LS	Low-Shapiro steady system	Simple and efficient; no root-finding; satisfying accuracy for reflections and transmissions
HR–S	Full subcritical steady system	Exactly well-balanced; most accurate hydrostatic reconstruction technique; requires a small nonlinear solve per interface per time step

This comparison indicates a structured trade-off. HR–S is described as the “gold-standard” well-balanced scheme for subcritical equilibria, whereas HR–LS is recommended for large-scale arterial network simulations because it remains simple and computationally less expensive while preserving low-Shapiro states and wave interactions with satisfying accuracy (Ghigo et al., 2017).

4. Shallow-water generalizations

In shallow-water research, generalized HR has expanded in at least three directions represented in the cited papers: entropy-stable flux correction, fully well-balanced hydrodynamic reconstruction for moving steady states, and entropy-stable path-conservative HR for one-layer and multilayer systems (Kivva, 2023, Berthon et al., 7 Jan 2025, Ersing et al., 2024).

In the entropy-stable flux-correction approach, the first-order HR scheme is retained as the low-order well-balanced, positivity-preserving baseline, while a hybrid interface flux is formed as

$A_0(x)$7

Here $A_0(x)$8 is the Rusanov flux and $A_0(x)$9 is any consistent high-order flux. The limiter $K(x)$0 is chosen by maximizing admitted antidiffusion subject to well-balancedness, positivity, and a fully discrete cell entropy inequality based on the physical entropy

$K(x)$1

and entropy flux

$K(x)$2

The paper formulates a large but linear program for the $K(x)$3, then derives explicit local upper bounds $K(x)$4, $K(x)$5, and $K(x)$6, setting

$K(x)$7

so that no global LP solver is needed (Kivva, 2023).

A different generalization is the fully well-balanced hydrodynamic reconstruction for shallow water over non-flat topography. The steady states satisfy

$K(x)$8

so exact preservation of moving steady states requires handling a nonlinear Bernoulli relation. The method avoids solving that nonlinear equation directly by introducing a correction $K(x)$9 into a hydrostatic-type reconstruction. After linearization, the final explicit formula for $$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$0 is given in Proposition 4.2, and the reconstructed heights become

$$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$1

If $$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$2, the correction vanishes and the standard lake-at-rest reconstruction is recovered; if the left and right states form a moving steady-state jump, then $$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$3 (Berthon et al., 7 Jan 2025).

For one-layer and multilayer shallow-water systems in path-conservative form,

$$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$4

another generalized HR constructs reconstructed bottom data and layer thicknesses so that the nonconservative volume terms cancel and the lake-at-rest is preserved for both SWE and ML-SWE. The interface bottom is chosen by

$$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$5

followed by reconstructed cumulative depths $$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$6 and layer thicknesses $$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$7, leading to interface states $$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$8 and $$(A_{L,R}^*,Q_{L,R}^*)=\mathcal R^*(A_{L,R},Q_{L,R},K_{L,R},Z_{L,R}),$$9 used in the finite-volume update (Ersing et al., 2024).

5. Stability, positivity, and high-order structure

A recurring feature of generalized HR is that well-balancedness is coupled to additional structural properties rather than treated in isolation. In the first-order shallow-water HR with flux correction, the baseline Rusanov HR scheme is stated to be well-balanced and positivity-preserving, to satisfy a semi-discrete in-cell entropy inequality, and to be highly diffusive; the high-order correction is therefore limited by constraints derived from positivity, hydrostatic preservation, and the discrete entropy inequality (Kivva, 2023).

In the path-conservative entropy-stable framework for SWE and ML-SWE, the entropy pair for SWE is

$$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$0

An entropy-conservative numerical flux satisfies

$$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$1

and entropy stability is obtained by adding LLF dissipation,

$$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$2

For the first-order HR finite-volume scheme, positivity preservation is stated under the CFL condition

$$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$3

and the same reconstruction is extended to a collocated nodal split-form discontinuous Galerkin spectral element method on curvilinear meshes, blended with a compatible subcell finite-volume method and an additional positivity-limiter at wet/dry fronts (Ersing et al., 2024).

High-order generalization also appears in the fully well-balanced hydrodynamic reconstruction. A local polynomial reconstruction of order $$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$4 is combined with a smoothness indicator

$$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$5

where $$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$6 measures the local departure from steady-state relations. The interface values and source term are then blended so that the resulting Runge–Kutta discretization is $$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$7th-order accurate away from steady states, preserves positivity of $$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$8, and exactly preserves both lake-at-rest and moving steady states (Berthon et al., 7 Jan 2025).

A further variant is the convex-combination reconstruction for shallow water with bounded velocities. There the reconstructed depth is defined by

$$\mathcal F^* = \mathcal F(U_L^*,U_R^*;K^*) + \text{source corrections},$$9

blending a depth-based reconstruction with a surface-based reconstruction. The weight $\mathcal R^*$0 is selected from a positivity indicator $\mathcal R^*$1 so that the reconstruction is positivity preserving and “self-monotone,” while a separate factor $\mathcal R^*$2 reduces the discharge slope when $\mathcal R^*$3, keeping reconstructed velocities bounded (Skevington, 2021). This suggests that generalized HR has become a broader design space in which interface reconstruction is adapted not only to source balance but also to entropy control, wet/dry behavior, and bounded-velocity requirements.

6. Applications, performance, and limitations

The blood-flow study provides the clearest direct performance comparison among generalized HR variants. In single-artery steady-state tests with smooth stenosis and discontinuous step changes, classical HR fails for low-Shapiro steady states: large $\mathcal R^*$4 or large jump amplitudes $\mathcal R^*$5 produce $\mathcal R^*$6 errors. HR–LS recovers low-Shapiro steady states with $\mathcal R^*$7 errors $\mathcal R^*$8, first-order convergence, and robustness to large jumps. HR–S is machine-exact well-balanced for all subcritical steady states (Ghigo et al., 2017).

In linear and nonlinear wave-reflection tests at a step, HR mis-estimates reflection and transmission for large jumps, whereas HR–LS and HR–S both capture correct amplitudes; HR–S is stated to be slightly more accurate but more expensive (Ghigo et al., 2017). In a 55-artery network with a pathological iliac stenosis, HR–LS and HR–S produce virtually identical pressure–flow waveforms at physiologically relevant monitoring sites, while HR deviates significantly. With added viscosity and viscoelastic wall damping, HR–LS and HR–S remain indistinguishable and HR remains poor. The reported computational cost is HR–LS $\mathcal R^*$9 classical HR, with HR–S about $$\frac{\partial A}{\partial t} +\frac{\partial Q}{\partial x} =0, \qquad \frac{\partial Q}{\partial t} +\frac{\partial}{\partial x}\Bigl(\frac{Q^2}{A}+\frac{K}{3\rho}\,A^{3/2}\Bigr) =\frac{A}{\rho}\Bigl[\partial_x(K\sqrt{A_0})-\tfrac{2}{3}\sqrt{A}\,\partial_xK\Bigr].$$0–$$\frac{\partial A}{\partial t} +\frac{\partial Q}{\partial x} =0, \qquad \frac{\partial Q}{\partial t} +\frac{\partial}{\partial x}\Bigl(\frac{Q^2}{A}+\frac{K}{3\rho}\,A^{3/2}\Bigr) =\frac{A}{\rho}\Bigl[\partial_x(K\sqrt{A_0})-\tfrac{2}{3}\sqrt{A}\,\partial_xK\Bigr].$$1 slower due to nonlinear solves, and all schemes remain stable under the standard CFL (Ghigo et al., 2017).

The shallow-water literature emphasizes different limitations. The first-order HR scheme is described as highly diffusive, whereas naive high-order HR variants may lose entropy stability and produce non-physical oscillations; the entropy-stable flux-correction strategy addresses this by allowing only the maximal antidiffusion consistent with well-balancedness, positivity, and the discrete entropy inequality (Kivva, 2023). The hydrodynamic reconstruction for moving steady states addresses the nonlinear Bernoulli difficulty without iterative solution, preserving both moving and non-moving steady solutions while reducing to classical hydrostatic reconstruction when the velocity vanishes and remaining consistent with the homogeneous flat-bottom system (Berthon et al., 7 Jan 2025).

A broader interpretation follows from these results. Generalized hydrostatic reconstruction is best understood as a systematic replacement of rest-state reconstruction by target steady-state reconstruction. In blood flow this target is low-Shapiro or fully subcritical equilibrium (Ghigo et al., 2017). In shallow water it may be lake-at-rest, moving Bernoulli equilibria, or entropy-stable path-conservative steady states on wet/dry and multilayer configurations (Berthon et al., 7 Jan 2025, Ersing et al., 2024). Across these settings, the central objective remains the same: to construct interface states and source corrections such that equilibrium preservation, positivity, and physically relevant wave dynamics are recovered without abandoning conservative finite-volume or DGSEM structure.