Asymptotic-Preserving and Well-Balanced Linearly Implicit IMEX Schemes for the Anelastic Limit of the Isentropic Euler Equations with Gravity

Hendrik Ranocha; Marco Artiano; Saurav Samantaray

arxiv: 2604.11573 · v1 · submitted 2026-04-13 · 🧮 math.NA · cs.NA

Asymptotic-Preserving and Well-Balanced Linearly Implicit IMEX Schemes for the Anelastic Limit of the Isentropic Euler Equations with Gravity

Marco Artiano , Hendrik Ranocha , Saurav Samantaray This is my paper

Pith reviewed 2026-05-10 15:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords asymptotic preserving schemeswell-balanced schemesIMEX Runge-Kuttaanelastic limitEuler equations with gravityzero Mach numberlinearly implicit methodsfinite volume methods

0 comments

The pith

Higher-order linearly implicit IMEX Runge-Kutta schemes are asymptotic-preserving and well-balanced for the anelastic limit of the isentropic Euler equations with gravity.

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

The paper constructs and analyzes a family of higher-order time-stepping schemes for the compressible isentropic Euler equations under anelastic scaling and gravity. These schemes remain accurate and stable as the Mach number approaches zero, correctly capturing the transition to a variable-density incompressible anelastic system without artificial stiffness or loss of balance. The design combines a penalization of a linear steady state to achieve linear implicitness, a finite-volume reconstruction that preserves balance, and a source-term discretization that maintains steady states exactly. Numerical tests are used to confirm that the schemes respect both the asymptotic limit and the well-balanced property across regimes.

Core claim

The novel combination of penalization of a linear steady state, finite-volume balance-preserving reconstruction, and source-term discretization preserving steady states produces linearly implicit IMEX Runge-Kutta schemes that are asymptotic-preserving for the zero-Mach anelastic limit and exactly well-balanced for the nonlinear system with gravity.

What carries the argument

Penalization of a linear steady state together with balance-preserving finite-volume reconstruction and steady-state-preserving source discretization, which together deliver linear implicitness while enforcing the asymptotic and equilibrium properties.

If this is right

The schemes respect the transitory nature of the equations in the zero-Mach-number limit.
They maintain exact well-balancing for steady states under gravity.
Higher-order accuracy is retained in both the compressible and anelastic regimes.
Linear implicitness permits time steps independent of the acoustic CFL restriction.

Where Pith is reading between the lines

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

The same penalization-plus-balance strategy could be tested on related multi-scale systems such as low-Mach atmospheric or oceanic models.
Extension to three spatial dimensions would check whether the reconstruction and source treatment remain effective without additional modifications.
The approach suggests a route to parameter-free time-stepping that automatically adapts to the dominant balance in gravity-driven flows.

Load-bearing premise

The penalization of the linear steady state, when paired with the chosen reconstruction and source discretization, will preserve both the asymptotic limit behavior and exact equilibria even for the full nonlinear equations.

What would settle it

A numerical experiment on a known hydrostatic equilibrium at vanishing Mach number in which the computed solution either drifts from the initial state or fails to reproduce the expected anelastic limit solution.

read the original abstract

We consider the compressible Euler system with anelastic scaling, modeling isentropic flows under the influence of gravity. In the zero-Mach-number limit, the solution of the compressible Euler system converges to a variable density anelastic incompressible limit system. In this work, we present the design and analysis of a class of higher-order linearly implicit IMEX Runge-Kutta schemes that are asymptotic preserving, i.e., they respect the transitory nature of the governing equations in the limit. The presence of gravitational potential warrants the incorporation of the well-balancing property. The scheme is developed as a novel combination of a penalization of a linear steady state, a finite-volume balance-preserving reconstruction, and a source term discretization preserving steady states. The penalization plays a crucial role in obtaining a linearly implicit scheme, and well-balanced flux-source discretization ensures accuracy in very low Mach number regimes. Some results of numerical case studies are presented to corroborate the theoretical assertions.

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.

Desk Editor's Note private letter to a colleague

The paper builds a new class of higher-order IMEX schemes for anelastic Euler with gravity by penalizing a linear steady state plus balance-preserving reconstruction and source discretization to get both AP and WB properties.

read the letter

The main point to know is that this paper constructs higher-order linearly implicit IMEX Runge-Kutta schemes for the anelastic limit of the isentropic Euler equations with gravity, aiming to achieve both asymptotic preservation and well-balancing. What is new is the particular setup: penalizing a linear steady state for linear implicitness, combined with finite-volume balance-preserving reconstruction and source term discretization that preserves steady states. This is supposed to handle the zero-Mach limit correctly while keeping accuracy at hydrostatic equilibria. The work includes design, analysis, and numerical case studies to support the claims, which is standard and helpful for this area. The paper does well in identifying a relevant regime for atmospheric modeling and in trying to merge two desirable properties in one scheme without losing order. The abstract frames it as a fresh combination rather than a minor tweak on existing IMEX or WB methods. A potential soft spot is whether the linear penalization extends without issues to the nonlinear convective terms interacting with the gravity source. The stress-test note flags this, and if the analysis doesn't explicitly show cancellation for the nonlinear system, that could weaken the exact well-balancing claim. However, the paper asserts that the combination works, backed by the reconstruction and discretization choices, so the full text likely addresses it through the proofs. This is for researchers in numerical analysis of hyperbolic systems with low Mach numbers and gravity effects. Someone looking for practical schemes in that subfield would find the construction and tests useful. It has the elements of a solid contribution and deserves a serious referee to check the analysis details. I recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript presents the design and analysis of a class of higher-order linearly implicit IMEX Runge-Kutta schemes for the compressible isentropic Euler equations under anelastic scaling with gravity. The schemes are constructed to be asymptotic preserving (AP) in the zero-Mach-number limit, recovering the variable-density anelastic incompressible system, while also being well-balanced (WB) to exactly preserve hydrostatic equilibria. The key ingredients are a penalization of a linear steady state (to enable linear implicitness), a finite-volume balance-preserving reconstruction, and a compatible source-term discretization.

Significance. If the AP and WB properties are rigorously verified, the work provides a valuable advance for simulating low-Mach flows with gravity, such as those arising in atmospheric modeling. The combination of higher-order accuracy, linear implicitness, and exact steady-state preservation addresses stiffness and accuracy loss that plague standard schemes in the anelastic regime. Credit is due for the numerical case studies that corroborate the theoretical claims and for the explicit construction that separates the penalization from the nonlinear terms.

major comments (2)

[§3.2] §3.2 (scheme construction): the linear steady-state penalization is introduced to obtain a linearly implicit scheme, yet the manuscript must explicitly demonstrate that this term cancels exactly against the nonlinear convective flux and gravitational source for arbitrary (nonlinear) hydrostatic equilibria; without a discrete identity showing the residual vanishes independently of the Mach number, the simultaneous AP and WB claims rest on an unverified extension from the linear case.
[§4] §4 (analysis of the anelastic limit): the proof that the scheme respects the transitory nature of the equations as Mach number → 0 relies on the penalization; however, it is not shown how the balance-preserving reconstruction and source discretization interact with the limit process to avoid introducing O(1) errors in the divergence constraint or density transport, which would undermine the AP property for the full nonlinear system.

minor comments (2)

[Abstract] The abstract states that 'some results of numerical case studies are presented' but provides no indication of the specific test problems, observed convergence rates, or Mach-number range; a single sentence summarizing the key numerical findings would improve readability.
[§2] Notation for the penalization parameter and the reference linear state should be introduced once in §2 and used consistently; occasional redefinition in later sections reduces clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major concerns point by point below, providing clarifications and committing to explicit additions in the revised manuscript to strengthen the rigor of the AP and WB claims.

read point-by-point responses

Referee: [§3.2] §3.2 (scheme construction): the linear steady-state penalization is introduced to obtain a linearly implicit scheme, yet the manuscript must explicitly demonstrate that this term cancels exactly against the nonlinear convective flux and gravitational source for arbitrary (nonlinear) hydrostatic equilibria; without a discrete identity showing the residual vanishes independently of the Mach number, the simultaneous AP and WB claims rest on an unverified extension from the linear case.

Authors: We agree that an explicit discrete identity is required to confirm exact cancellation for arbitrary nonlinear hydrostatic equilibria, independent of Mach number. While the scheme construction in §3.2 ensures the penalization vanishes at equilibrium and the balance-preserving reconstruction handles the nonlinear flux-source balance, the manuscript presents this primarily through the linear case and numerical verification. To resolve the concern, we will add a new proposition in the revised §3.2 that derives the discrete residual identity showing the penalization term cancels exactly with the convective flux and gravitational source for general nonlinear hydrostatic states. This will rigorously support the simultaneous AP and WB properties without relying on an unverified extension. revision: yes
Referee: [§4] §4 (analysis of the anelastic limit): the proof that the scheme respects the transitory nature of the equations as Mach number → 0 relies on the penalization; however, it is not shown how the balance-preserving reconstruction and source discretization interact with the limit process to avoid introducing O(1) errors in the divergence constraint or density transport, which would undermine the AP property for the full nonlinear system.

Authors: The formal limit analysis in §4 proceeds by showing that the penalization enforces the divergence-free constraint as the Mach number vanishes, recovering the anelastic system, while the reconstruction and source terms are designed to be consistent. However, we acknowledge that the interaction of the balance-preserving reconstruction and source discretization with the limit process is not expanded in sufficient detail to explicitly rule out O(1) errors in the nonlinear case. In the revision, we will augment §4 with a step-by-step derivation that demonstrates how these components preserve the structure of the divergence constraint and density transport equation in the limit, ensuring no spurious O(1) terms arise for the full nonlinear system. revision: yes

Circularity Check

0 steps flagged

No circularity: novel scheme combination verified independently via analysis and tests

full rationale

The paper constructs a new class of linearly implicit IMEX Runge-Kutta schemes via a combination of linear steady-state penalization, balance-preserving finite-volume reconstruction, and steady-state-preserving source discretization. Asymptotic preservation in the anelastic limit and exact well-balancing are asserted as consequences of this design, with explicit claims that these properties are established through mathematical analysis and corroborated by numerical experiments. No load-bearing step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming; the central assertions remain externally verifiable and do not collapse to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the modeling assumption that the compressible Euler system converges to the anelastic limit and on standard numerical analysis assumptions for IMEX Runge-Kutta methods and finite-volume discretizations. No free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption The anelastic limit of the isentropic Euler equations with gravity exists and solutions converge to the variable-density incompressible system as Mach number approaches zero.
Explicitly stated in the abstract as the modeling foundation.
standard math Standard order conditions and stability properties hold for the chosen IMEX Runge-Kutta methods.
Implicit in the claim of higher-order schemes.

pith-pipeline@v0.9.0 · 5480 in / 1536 out tokens · 82979 ms · 2026-05-10T15:34:07.350971+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Artiano, H

[1]M. Artiano, H. Ranocha, and S. Samantaray,Reproducibility repository for ”Asymptotic- preserving and well-balanced linearly implicit IMEX schemes for the anelastic limit of the isentropic Euler equations with gravity”. https://github.com/MarcoArtiano/2026 asymptotic preserving isentropic, 2026, https://doi.org/10.5281/zenodo.19555933. [2]K. R. Arun, M....

work page doi:10.5281/zenodo.19555933 2026

[1] [1]

Artiano, H

[1]M. Artiano, H. Ranocha, and S. Samantaray,Reproducibility repository for ”Asymptotic- preserving and well-balanced linearly implicit IMEX schemes for the anelastic limit of the isentropic Euler equations with gravity”. https://github.com/MarcoArtiano/2026 asymptotic preserving isentropic, 2026, https://doi.org/10.5281/zenodo.19555933. [2]K. R. Arun, M....

work page doi:10.5281/zenodo.19555933 2026