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arxiv: 2604.23780 · v1 · submitted 2026-04-26 · 🧮 math.NA · cs.NA

Asymptotic preserving scheme for the shallow water equations with non-flat bottom topography and Manning friction term

Guanlan Huang , Sebastiano Boscarino , Tao Xiong This is my paper

Pith reviewed 2026-05-08 05:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords asymptotic preserving schemesshallow water equationsManning frictionwell-balanced schemessemi-implicit IMEX-RKWENO schemesfinite difference methods
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The pith

A high-order scheme for shallow water equations preserves asymptotic properties without penalization by implicit treatment of friction.

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

The paper develops a class of high-order asymptotic preserving finite difference schemes for the shallow water equations that include non-flat bottom topography and Manning friction. By removing the penalization term used in earlier work and instead using a semi-implicit IMEX-RK time discretization with WENO for first derivatives and central differences for second derivatives, the new approach maintains the AP, asymptotically accurate and well-balanced properties while reducing computational cost. A reader would care because it enables more efficient simulations of flows where friction effects vary, such as in rivers or floods, without sacrificing the ability to handle different physical regimes accurately.

Core claim

By employing a high order semi-implicit IMEX-RK time discretization coupled with high-order WENO reconstruction for first-order derivatives and central difference for second-order derivatives, the schemes achieve fully high-order accuracy while preserving the asymptotic preserving property without the need for penalization, leading to higher efficiency especially in the intermediate regime between convection and diffusion. Treating the momentum in the friction terms implicitly is essential for preserving the AP property; otherwise the scheme fails to converge to the limiting equations.

What carries the argument

High-order semi-implicit implicit-explicit Runge-Kutta (SI-IMEX-RK) time discretization with WENO reconstruction for first derivatives, central differences for second derivatives, and implicit treatment of Manning friction in the momentum equation.

If this is right

  • The schemes retain AP, asymptotically accurate and well-balanced properties.
  • They offer higher computational efficiency than previous penalized schemes especially in the intermediate regime.
  • Implicit treatment of Manning friction is essential to preserve the AP property and stability.

Where Pith is reading between the lines

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

  • The implicit friction treatment could extend to other hyperbolic balance laws with stiff source terms for similar efficiency gains.
  • Real-world tests on river or coastal flow data might reveal practical speedups without accuracy loss.

Load-bearing premise

That the high-order semi-implicit IMEX-RK discretization with WENO and central differences will preserve the asymptotic preserving property and higher efficiency without the penalization term used in prior work.

What would settle it

Numerical test where the scheme with explicit friction treatment fails to converge to the limiting diffusion equations as the friction coefficient grows large, or where the new scheme shows no efficiency gain over penalized versions in intermediate convection-diffusion regimes.

Figures

Figures reproduced from arXiv: 2604.23780 by Guanlan Huang, Sebastiano Boscarino, Tao Xiong.

Figure 4.1
Figure 4.1. Figure 4.1: Example 4.3, the numerical results obtained by SI-S1 for the water depth h (left) and the momentum m (right) for the smooth initial conditions (4.6) and the discontinuous initial conditions (4.7) with different ε. Top: smooth; Bottom: discontinuous (N = 400 uniform meshes). We choose the following well-prepared initial condittions: (4.9)    h(x, y, 0) = sin(π(x + y)) + 2, m1(x, y, 0) = −2π cos(π(x +… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Example 4.3, the numerical results about the water depth h (left) and momentum m (right) for the smooth initial conditions (4.6) with N = 400 uniform meshes view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Example 4.3, the numerical results about the water depth h (left) and the momentum m (right) for the discontinuous initial conditions (4.7) with N = 400 uniform meshes. Example 4.5. (2D accuracy test with nonlinear friction) In this example, we aim to test the accuracy of the two schemes in a nonlinear two-dimensional setting. The well-prepared initial conditions are crucial for achieving the AA property… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Numerical solutions of the surface level h + b for Example 4.6 with 200 × 100 uniform meshes. From top to bottom: at t = 0.12 from 0.9998 to 1.0060; at t = 0.24 from 0.9967 to 1.0130; at t = 0.36 from 0.9901 to 1.0097; t = 0.48 from 0.9906 to 1.0043; t = 0.6 from 0.9955 to 1.0045. 30 contour lines are used. Left: SI-S1 scheme; Right: SI-S2 scheme view at source ↗
read the original abstract

In our previous work [29], we proposed a class of high-order asymptotic preserving (AP) finite difference weighted essentially non-oscillatory (WENO) schemes for solving the shallow water equations (SWEs) with bottom topography and Manning friction, utilizing a penalization technique inspired by [6]. Although the added weighted diffusive term enhanced stability, it increased computational cost and slowed down the convergence rate in the intermediate regime between convection and diffusion. In this paper, we extend our previous study by removing the penalization while preserving the AP property. To achieve this, we employ a high order semi-implicit implicit-explicit Runge-Kutta (SI-IMEX-RK) time discretization, coupled with the high-order WENO reconstruction for first-order derivatives and a central difference scheme for second-order spatial derivatives. This combination yields a class of fully high-order schemes. Theoretical analysis and numerical experiments demonstrate that the proposed schemes retain AP, asymptotically accurate (AA) and well-balanced properties, while offering higher computational efficiency compared to our previous schemes in [29], especially in the intermediate regime between convection and diffusion. Moreover, treating the momentum in the friction terms implicitly is essential for preserving the AP property; otherwise, the scheme fails to converge to the limiting equations. This indicates that implicit treatment of Manning friction is necessary for the stability of the method.

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 / 2 minor

Summary. The manuscript develops a high-order semi-implicit IMEX-RK finite-difference scheme for the shallow water equations with non-flat bottom topography and Manning friction. It removes the penalization term from the authors' prior work [29], employing WENO reconstruction for first derivatives and central differences for second derivatives. Theoretical analysis establishes consistency with the limiting diffusion equations in the appropriate regime, while numerical experiments confirm that the scheme retains the asymptotic preserving (AP), asymptotically accurate (AA), and well-balanced properties and achieves higher efficiency than the penalized predecessor, especially in the intermediate convection-diffusion regime. The analysis further shows that implicit treatment of momentum in the friction terms is required for the AP property; explicit treatment fails to converge to the limit.

Significance. If the derivations and experiments hold, the work provides a practical advance in asymptotic-preserving discretizations for balance laws with stiff nonlinear sources. Removing the penalization while preserving AP/AA/well-balanced behavior and improving efficiency in transitional regimes addresses a clear computational drawback of the earlier method. The direct verification that explicit friction treatment breaks the limit while the implicit version succeeds supplies a concrete, falsifiable insight applicable to related problems in shallow-water and related hyperbolic-parabolic systems.

minor comments (2)
  1. [Theoretical analysis] §3 (or the section presenting the limiting analysis): the consistency argument with the diffusion limit would benefit from an explicit statement of the precise scaling regime (e.g., the relation between the friction coefficient and the small parameter) under which the scheme is shown to be AP.
  2. [Numerical experiments] Numerical experiments section: the description of the fixed-point iteration used to solve the nonlinear implicit friction term should include the convergence tolerance and the maximum number of iterations employed, to allow readers to assess the practical cost.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately captures the main contributions: the removal of the penalization term from our prior work, the use of SI-IMEX-RK time discretization with WENO for first derivatives and central differences for second derivatives, the theoretical consistency with the limiting diffusion equations, and the numerical confirmation of AP, AA, and well-balanced properties along with improved efficiency, particularly in the intermediate regime. We also note the referee's observation that implicit treatment of the momentum in the friction terms is required for the AP property.

Circularity Check

0 steps flagged

Minor self-citation to prior schemes; new AP analysis and verification are independent

full rationale

The paper references prior work [29] only for comparison of efficiency and to motivate removal of the penalization term. The central claims rest on new theoretical analysis showing consistency with the diffusion limit under the SI-IMEX-RK discretization, plus explicit numerical checks that explicit friction treatment fails to reach the limit while the implicit version succeeds. These verifications are presented as direct evidence rather than fitted quantities or self-referential definitions. No load-bearing step reduces by construction to the inputs or to an unverified self-citation chain. The well-balanced property is maintained via flux differencing that cancels topography terms at steady state, an independent structural property.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard well-posedness assumptions for the shallow water equations and on the unproven claim that the chosen discretization preserves the asymptotic limit; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The shallow water equations with bottom topography and Manning friction admit a well-defined asymptotic limit under standard smoothness assumptions.
    Invoked implicitly as background for the AP property.
  • ad hoc to paper The SI-IMEX-RK scheme with implicit momentum friction treatment converges to the limiting equations.
    Central claim of the paper; stated as necessary but not derived in the abstract.

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