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arxiv: 2605.02560 · v1 · submitted 2026-05-04 · 🧮 math.NA · cs.NA· physics.comp-ph

Well-Balanced Subcell Limiting for Discontinuous Galerkin Discretizations of the Shallow-Water Equations

Andr\'es M. Rueda-Ram\'irez , Patrick Ersing , Andrew R. Winters , Gregor J. Gassner This is my paper

Pith reviewed 2026-05-08 19:15 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords shallow water equationsdiscontinuous Galerkinwell-balanced methodssubcell limitingfinite volume limitersnon-conservative balance lawsequilibrium preservation
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The pith

Reformulating the shallow water equations allows staggered fluxes that stay exactly balanced under node-wise DG limiting.

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

The paper addresses the loss of well-balanced properties when high-order discontinuous Galerkin methods are combined with subcell finite-volume limiters for the shallow water equations over variable topography. Standard flux differencing breaks exact equilibrium preservation at nodes even when the separate DG and FV pieces are balanced. By rewriting the equations so the source term becomes proportional to the gradient of total water height, the authors build staggered fluxes whose non-conservative terms take a local-times-jump form and cancel individually at steady state. This permits arbitrary nodal blending with low-order fluxes while keeping high-order accuracy and robustness. The result is a hybrid scheme that handles shocks, wet-dry fronts, and complex obstacles without destroying equilibrium states.

Core claim

The central claim is that a novel flux-differencing formulation for non-conservative systems constructs staggered DG fluxes whose non-conservative contributions vanish individually at equilibrium. This is achieved by introducing a reformulation of the shallow water equations in which the source term is proportional to the gradient of the total water height, allowing the design of fluxes that preserve equilibrium locally at the node level and thereby enable arbitrary nodal blending with low-order FV fluxes. The resulting DG/FV method is high-order accurate, robust, and exactly well-balanced under node-wise limiting, and the same construction applies to a broader class of nonlinear balance-law

What carries the argument

Staggered DG fluxes in local-times-jump form for non-conservative terms, built from the reformulation that makes the source term proportional to the gradient of total water height.

If this is right

  • The hybrid method retains high-order accuracy under node-wise limiting.
  • Steady states with variable bottom topography are preserved exactly at the discrete nodes.
  • The scheme remains robust for dam-break flows, wet-dry fronts, and obstacle interactions.
  • The construction extends to other nonlinear systems of balance laws.

Where Pith is reading between the lines

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

  • Practical flood and coastal models could use fewer special equilibrium fixes.
  • Similar staggered flux constructions may apply to other non-conservative hyperbolic systems such as those with gravity or chemistry source terms.
  • The node-level exactness could simplify adaptive mesh refinement or parallel implementations.

Load-bearing premise

The reformulation of the shallow water equations in which the source term is proportional to the gradient of the total water height allows construction of staggered fluxes that preserve equilibrium locally at the node level.

What would settle it

A lake-at-rest test on non-flat topography in which the limited scheme produces non-zero velocities or height changes after many time steps.

Figures

Figures reproduced from arXiv: 2605.02560 by Andr\'es M. Rueda-Ram\'irez, Andrew R. Winters, Gregor J. Gassner, Patrick Ersing.

Figure 1
Figure 1. Figure 1: Two different labeling possibilities for the Legendre-Gauss-Lobatto nodes in one dimension. 𝚪 𝑎 (𝑗,𝑘) = min( ∑ 𝑗,𝑘) 𝑙=0 ∑ 𝑁 𝑚=0 𝑆𝑙𝑚𝐟 ∗ (𝑙,𝑚) + 𝚽loc 𝑗 ◦ min( ∑ 𝑗,𝑘) 𝑙=0 ∑ 𝑁 𝑚=0 𝑆𝑙𝑚𝚽 jump (𝑙,𝑚) , ∀(𝑗, 𝑘) ⧵ {(0,−1),(𝑁, 𝑁 + 1),(𝑁, 𝑁 − 1)}, (28) 𝚪 𝑎 (𝑁,𝑁−1) = 𝑁 ∑ −1 𝑙=0 ∑ 𝑁 𝑚=0 𝑆𝑙𝑚𝐟 ∗ (𝑙,𝑚) + 𝚽loc 𝑁 ◦ 𝑁 ∑ −1 𝑙=0 ∑ 𝑁 𝑚=0 𝑆𝑙𝑚𝚽 jump (𝑙,𝑚) +2𝚽loc 𝑁 ◦𝚽 jump (𝑁,0), (29) 𝚪 𝑎 (𝑁,𝑁+1) = ̂𝐟 (𝑁,𝑅) + 𝚽 ◊ (𝑁,𝑅) . (30) Repea… view at source ↗
Figure 2
Figure 2. Figure 2: Curvilinear mesh (a) and random node-wise blending coefficients (b) used for the well-balancedness test. (a) The well-balanced scheme introduced in Section 3, which combines the novel flux-differencing formulation with the reformulated fluxes of Ersing et al. [26] (here denoted as Ersing-jump). The corresponding two-dimensional fluxes and non-conservative terms are detailed in Appendix B, but are reproduce… view at source ↗
Figure 3
Figure 3. Figure 3: Contour plots showing the lake-at-rest error 𝐻(⃗𝑥, 𝑡) − 𝐻(⃗𝑥, 0) at time 𝑡 = 10 for three different configurations of fluxes and flux-differencing formulas. The color range is adjusted to the solution range in each subplot. 𝐟 1∗(𝐮𝐿, 𝐮𝑅) = ⎛ ⎜ ⎜ ⎜ ⎝ {{ℎ𝑣1 }} {{ℎ𝑣1 }} {{𝑣1 }} + 𝑔 2 ℎ𝐿ℎ𝑅 {{ℎ𝑣1 }} {{𝑣2 }} ⎞ ⎟ ⎟ ⎟ ⎠ , 𝚽̃ 1⋆ ( 𝐮𝐿, 𝐮𝑅, ( 𝐽 ⃗𝑎1 ) 𝐿 , ( 𝐽 ⃗𝑎1 ) 𝑅 ) = ⎛ ⎜ ⎜ ⎜ ⎝ 0 𝑔ℎ𝐿 𝑔ℎ𝐿 ⎞ ⎟ ⎟ ⎟ ⎠ ◦ ⎛ ⎜ ⎜ ⎜ ⎝ 0 {{𝐽 … view at source ↗
Figure 4
Figure 4. Figure 4: Three-dimensional visualization of the circular dam break simulation after 20 time steps for the two different staggered flux formulas considered in this work view at source ↗
Figure 5
Figure 5. Figure 5: (top) Physical domain and cross section for channel flow past an isolated obstacle. (bottom) Decomposition of physical domain into non-overlapping, linear quadrilateral elements. desingularization formula from [56] to adjust the momentum at each node or set the momentum to zero at dry nodes ℎ𝑣1,2 = ⎧ ⎪ ⎨ ⎪ ⎩ 2ℎ 2ℎ𝑣1,2 ℎ2+max(ℎ2,𝜏𝑣𝑒𝑙) if ℎ > 5𝜖𝑛𝑢𝑚, 0 otherwise, (47) where 𝜖𝑛𝑢𝑚 = 10−13 is the tolerance for d… view at source ↗
Figure 6
Figure 6. Figure 6: Solution (left) and limiting coefficients (right) for the dam break flow past an obstacle at 𝑡 = 10 with polynomial degree 𝑁 = 3 in each spatial direction in each element. The node-wise limiting concentrates dissipation near complex flow features view at source ↗
Figure 7
Figure 7. Figure 7: Solution (left) and limiting coefficients (right) for the dam break flow past an obstacle at 𝑡 = 10 with polynomial degree 𝑁 = 5 in each spatial direction in each element. The node-wise limiting concentrates dissipation near complex flow features. divergence of a quantity that vanishes at equilibrium. This reformulation enables the construction of compatible two￾point fluxes based on the jump of that quant… view at source ↗
Figure 8
Figure 8. Figure 8: Water height histories at gauges G2, G4, G5, and G6. (grant agreement No. 101167322 - TRANSDIFFUSE). Gregor J. Gassner and Andrés M. Rueda-Ramírez acknowledge funding through the German Federal Ministry for Education and Research (BMBF) project “ICON-DG” (01LK2315B) of the “WarmWorld Smarter” program. CRediT authorship contribution statement Andrés M. Rueda-Ramírez: Conceptualization, Formal analysis, Meth… view at source ↗
read the original abstract

High-order discontinuous Galerkin (DG) methods equipped with subcell finite-volume (FV) limiters provide an efficient framework for the simulation of nonlinear hyperbolic balance laws featuring shocks and complex flow structures. However, for systems with non-conservative terms, the design of hybrid DG/FV schemes that simultaneously guarantee high-order accuracy, robustness, and well-balancedness remains challenging. In particular, for the shallow water equations with variable bottom topography, standard flux-differencing formulations combined with node-wise subcell limiting generally destroy the well-balanced property, even if both the underlying DG and FV methods are individually well-balanced. In this work, we propose a novel flux-differencing formulation for non-conservative systems that enables node-wise subcell limiting while preserving steady states exactly. The key idea is to construct staggered DG fluxes whose non-conservative contributions are in local-times-jump form and vanish individually at equilibrium. To achieve this, we introduce a reformulation of the shallow water equations in which the source term is proportional to the gradient of the total water height. This reformulation allows the design of staggered fluxes that preserve equilibrium locally at the node level, thereby enabling arbitrary nodal blending with low-order FV fluxes. The resulting DG/FV method is high-order accurate, robust, and exactly well-balanced under node-wise limiting. Numerical experiments, including two-dimensional dam-break configurations with wet/dry fronts and complex obstacle interactions, demonstrate the improved stability and accuracy of the proposed approach. Although this work focuses on the shallow water equations, the well-balanced hybrid DG/FV methods developed here are applicable to a broader class of nonlinear systems of balance laws.

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

0 major / 4 minor

Summary. The manuscript proposes a reformulation of the shallow-water equations in which the source term is made proportional to the gradient of the total height H = h + b. This permits construction of staggered DG fluxes whose non-conservative contributions appear in local-times-jump form and therefore vanish nodewise at equilibrium states (constant H, u = 0). The resulting hybrid DG/FV scheme with arbitrary node-wise subcell limiting is shown to remain exactly well-balanced while retaining high-order accuracy and robustness. Numerical experiments on two-dimensional dam-break problems with wet/dry fronts and obstacle interactions are presented to illustrate the properties.

Significance. If the exact well-balanced property under arbitrary nodal blending holds, the work resolves a recognized difficulty in hybrid high-order schemes for balance laws: standard flux-differencing formulations lose equilibrium preservation once subcell limiting is introduced. The reformulation and staggered-flux construction supply a parameter-free mechanism that preserves the steady-state cancellation locally at each node, independent of the blending weights. This is a concrete advance for long-time simulations over variable topography and extends in principle to other non-conservative balance laws.

minor comments (4)
  1. Abstract: the phrase 'local-times-jump form' is used without a parenthetical reference to the equation or subsection where the form is defined; adding such a pointer would improve immediate readability for readers unfamiliar with the construction.
  2. Section 3 (flux construction): the equivalence between the reformulated system and the original shallow-water equations is asserted but the algebraic steps that recover the standard momentum source term are not written out; a short displayed identity would remove any doubt.
  3. Numerical experiments: the order-of-accuracy study for the hybrid scheme on a smooth, non-equilibrium test case is not reported; inclusion of at least one such table (e.g., L2 errors versus polynomial degree) would directly support the high-order claim.
  4. Figure 4 (wet/dry dam-break): the color scale and contour levels are not stated in the caption; explicit mention of the plotted quantity (e.g., water height or total H) and the range would aid interpretation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The referee's summary correctly identifies the core contribution: a reformulation of the shallow-water equations that places the source term proportional to the gradient of total height H, enabling staggered DG fluxes whose non-conservative terms vanish nodewise at equilibrium and thereby permit arbitrary node-wise subcell limiting while preserving exact well-balancedness. We appreciate the recognition that this addresses a known difficulty in hybrid high-order schemes for balance laws.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins with an explicit reformulation of the shallow-water equations that makes the source term proportional to the gradient of total height H = h + b. This is used to construct staggered DG fluxes in local-times-jump form whose non-conservative contributions vanish nodewise at equilibrium (constant H, u = 0). The hybrid DG/FV update is then obtained by arbitrary nodal blending with a low-order well-balanced FV scheme; the equilibrium preservation follows directly from the local vanishing property and does not rely on any fitted parameter, self-referential definition, or load-bearing self-citation. The paper states that the reformulation is equivalent to the original SWE and demonstrates the property through explicit flux differencing and numerical tests. No step in the chain reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on a domain-specific reformulation of the source term and the assumption that staggered fluxes can be constructed to cancel locally; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Reformulation of the shallow water equations makes the source term proportional to the gradient of total water height.
    This step is presented as the enabling change that lets non-conservative contributions vanish at equilibrium.

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