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A MUSCL-Hancock scheme for non-local conservation laws

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Pith's one-line read This paper proposes and proves convergence of a MUSCL-Hancock-type second-order scheme for non-local conservation laws.

desk verdict Solid a priori analysis for a new MUSCL-Hancock scheme, but the entropy convergence theorem only covers a slope-capped variant that is not the scheme implemented or tested. read the letter →

arxiv 2506.04176 v1 pith:CL2MEWPA submitted 2025-06-04 math.NA cs.NA

Nikhil Manoj , G. D. Veerappa Gowda , Sudarshan Kumar K This is my paper
classification math.NAcs.NA MSC 65M0865M1235L6535R09
keywords MUSCL-Hancocknon-localconservationlawsfinitevolumemethodentropysolutionsecond-orderaccuracyconvergenceanalysisLax-Friedrichsfluxmesh-dependentslopelimiter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that a single-stage MUSCL-Hancock finite volume scheme, with a careful discretization of the convolution term, is second-order accurate and converges to the unique entropy solution of a general class of one-dimensional non-local conservation laws. This matters because first-order schemes for these equations are robust but diffusive, while higher-order methods for non-local problems rarely come with convergence proofs. The proof derives uniform $L^\infty$, bounded-variation, and $L^1$-time-continuity estimates, extracts a convergent subsequence by compactness, and then uses a mesh-dependent slope modification to show the limit is the entropy solution, not merely a weak solution. Numerical tests in the paper report second-order accuracy and better efficiency per unit error than a two-stage Runge-Kutta MUSCL scheme.

What carries the argument

The object that carries the argument is the MUSCL-Hancock update: a piecewise-linear reconstruction with minmod slopes, a Taylor-series predictor that advances interface values to the half time level $t^{n+\frac{1}{2}}$, and a corrector that evaluates a Lax-Friedrichs flux. The convolution $A=\mu\ast\rho$ is discretized so that the interface data remain second-order consistent: midpoint quadrature at cell centers, a central-difference correction for the left/right interface values, and trapezoidal quadrature for the half-time convolutions. The proof also relies on the mesh-dependent slope limiter (5.4), which caps the reconstructed slopes at $K(\Delta x)^\delta$; this cap is what makes the correction relative to the first-order scheme small enough to fit the entropy-transfer theorem.

What would settle it

On a sequence of meshes $\Delta x \to 0$ with the same initial datum, compare the scheme with modified slopes (5.4) against the scheme with the implemented slopes (3.2): if the correction term $e^{n+1}_{j+\frac{1}{2}}$ computed with (3.2) fails to be uniformly $O((\Delta x)^\delta)$, or if the two schemes' solutions differ in $L^1$ by more than $O((\Delta x)^\delta)$, then the entropy-convergence theorem does not cover the code that was actually run.

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Extended reading notes

Core claim

The central claim is Theorem 5.2: for nonnegative initial datum $\rho_0 \in L^\infty \cap BV(\mathbb{R};\mathbb{R}_+)$, if the CFL condition (4.12) holds, then the piecewise-constant approximations produced by the scheme (3.5) with the modified slopes (5.4) converge in $L^1_{\mathrm{loc}}$ to the unique entropy solution of (2.1). The proof rewrites the second-order update as a first-order Lax-Friedrichs update plus a correction term, bounds that correction uniformly by $K(\Delta x)^\delta$, and invokes an entropy-transfer theorem that turns such a small correction, together with the $L^\infty$, BV, and time-continuity estimates, into full entropy convergence. The same estimates also show, without the slope modification, that the scheme converges to a weak solution; the numerical experiments indicate that the implemented scheme attains the expected second order.

Load-bearing premise

The entropy proof analyzes slopes clipped by a fixed constant $K$, but the implemented slopes (3.2) are recovered only when the mesh is not too fine; the step from the analyzed scheme to the implemented scheme as $\Delta x \to 0$ is asserted in a remark, not proved.

Editorial extensions

If this is right

  • Within the problem class (2.1), users can replace a first-order Lax-Friedrichs scheme by this MUSCL-Hancock scheme and keep a convergence guarantee under the CFL condition (4.12).
  • The mesh-dependent slope modification gives a general recipe: any second-order scheme whose flux correction over a first-order base is $O((\Delta x)^\delta)$ inherits entropy convergence from Theorem 5.1.
  • Because the modified-slope scheme is only used for the proof and the practical slopes are recovered on every fixed mesh, the method is expected to converge to entropy-admissible shocks and rarefactions rather than to a spurious weak solution.
  • The single-stage predictor-corrector structure makes the method cheaper per time step than a two-stage Runge-Kutta MUSCL scheme, and the numerical section reports this as a lower $L^1$ error for the same CPU time.
  • Even if the slope modification were dropped, the compactness estimates in the paper still produce convergence of a subsequence to a weak solution, so the scheme is not unstable without the modification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • As $\Delta x$ shrinks past any fixed $K$, the modified slopes (5.4) stop coinciding with the implemented slopes (3.2), so the entropy convergence of the implemented scheme is an extrapolation from the analyzed scheme rather than a proved consequence.
  • The same quadrature-plus-correction template could be applied to other single-stage or two-stage MUSCL variants, including the Runge-Kutta MUSCL scheme that the paper compares numerically, if a matching $O((\Delta x)^\delta)$ correction bound can be verified.
  • A practical check of the proof's bridge would be to monitor $\|e^{n+1}_{j+\frac{1}{2}}\|_{L^\infty}$ on the implemented slopes over refining meshes; the paper's entropy argument requires this quantity to decay like a fixed power of the mesh size.
  • The final claim is conditional on the entropy-transfer theorem imported from earlier work, so a self-contained proof of that transfer result would remove the main external dependency.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes a single-stage MUSCL-Hancock-type second-order finite volume scheme for scalar non-local conservation laws of the form (2.1), with the convolution appearing in the flux. The scheme uses a Lax-Friedrichs-type numerical flux and carefully designed discrete convolutions in the predictor and corrector steps. The authors derive positivity preservation, L1-stability, an L∞ bound, a BV estimate, and an L1-Lipschitz time-continuity estimate under an explicit CFL condition (4.12), and use Kolmogorov compactness to obtain convergence to a weak solution. Entropy convergence is then claimed for a slope-modified variant (5.4) by writing the scheme as a perturbation of a first-order Lax-Friedrichs scheme and invoking a perturbation theorem (Theorem 5.1) from the authors' earlier work [35]. Numerical experiments for smooth and discontinuous data compare the method with a first-order scheme and a MUSCL-Runge-Kutta scheme.

Significance. If the entropy convergence were established for the scheme that is actually proposed and tested, this would be a valuable contribution: it would provide a provably entropy-stable second-order method for a general class of non-local conservation laws, with detailed a priori estimates and explicit constants. The paper's strengths include the careful construction of the discrete convolutions, the explicit CFL condition, the positivity-preservation and BV/time-continuity analysis, and the numerical comparison with a first-order scheme and a MUSCL-RK2 scheme. The perturbation framework of Theorem 5.1 is a useful structural tool. However, the main advertised result is proved only for a slope-capped variant, and the bridge to the implemented scheme is not valid in the convergence limit, so the central claim is currently not established for the method used in Section 6.

major comments (2)
  1. [§5 (Theorem 5.2 and Remark 5)] The entropy convergence theorem is proved only for the scheme (3.5) with the modified slopes (5.4), in which the limiter is capped by sgn(ρ_{j+1}-ρ_j) K (Δx)^δ for fixed K>0 and δ∈(0,1). The numerical experiments in Section 6, however, use the uncapped slopes (3.2). At a discontinuity these slopes are O(1), so the correction terms e^{n+1}_{j+1/2} in (5.3) are O(1), not O((Δx)^δ). Remark 5 attempts to bridge this gap by choosing K large for each mesh, but the required K grows like Δx^{-δ} and is therefore not independent of Δx, as Theorem 5.1(ii) explicitly demands. Consequently, Theorem 5.2 does not apply to the scheme actually implemented and tested, and Remark 6 only asserts weak-solution convergence for the uncapped scheme. The paper's central claim that the proposed MUSCL-Hancock scheme converges to the unique entropy solution is therefore not established for the method presented in Section 6. The authors should either prove entropy convergence for the uncapped slopes (3.2) by a different argument or implement and test the modified scheme (5.4) with a fixed K.
  2. [§5 (Lemma 5.1)] The proof of Lemma 5.1 defines the constant K in terms of itself. In the estimate (5.8), the bound on |ρ^{n+1/2,-}_{j+1/2}-ρ^n_j| contains the term (1/2 + λ/2 ‖∂_ρ f‖) K (Δx)^δ, and the constant ilde K is then defined with this K. After (5.10), the final constant is set to K = (λ‖∂_ρ f‖+α+2M‖μ‖L_μ) ilde K, which makes K appear on both sides. The existence of a fixed K is not established; one either needs to verify a fixed-point condition for the coefficient on K or, more simply, distinguish the slope-cap constant from the constant in the correction bound and show that a sufficiently large cap yields a uniform bound. This is a load-bearing point because Lemma 5.1 supplies hypothesis (ii) of Theorem 5.1.
minor comments (5)
  1. [Abstract and Conclusion] The abstract and the conclusion state that the scheme converges to the entropy solution without mentioning that this is proved only for the mesh-dependent slope modification (5.4), while the numerical experiments use the unmodified limiter (3.2). The statements should be qualified to match the theorems.
  2. [§5 (Theorem 5.1)] Theorem 5.1 is quoted from the authors' earlier paper [35] and its proof is not included; since it is a published result this is acceptable, but the manuscript should state explicitly which properties of the base scheme are inherited and perhaps include a proof or a precise reference to the corresponding theorem in [35].
  3. [§5 (Lemma 5.1)] In the estimate (5.8), the final inequality Δx ≤ (Δx)^δ requires Δx ≤ 1, which is not stated. Since the argument is intended for the limit Δx→0, the condition should be made explicit.
  4. [§6 (Numerical experiments)] The numerical experiments set α=0.16 and use Δt = Δx/20, but the text does not verify that this choice satisfies the CFL condition (4.12) for the flux functions and kernels used. A short check or statement would be helpful.
  5. [§6 (Examples)] Example 2 says 'the same setup as in Example 2' but clearly means Example 1; and the sentence before Theorem 5.2 says 'we are now in a position to use Theorem 5.2' where Theorem 5.1 is meant. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: entropy convergence is anchored in an external first-order LF scheme and a Vila-type perturbation lemma.

full rationale

The derivation chain is not circular. Theorem 5.2 establishes entropy convergence for the slope-capped scheme by writing (3.5) in the perturbative form (5.2)-(5.3), where the base scheme is the first-order Lax-Friedrichs scheme (3.15) whose convergence to the entropy solution is cited to external sources [6,3,1]. The perturbation lemma (Theorem 5.1) is quoted as originally established by Vila [37] and adapted in the authors' prior [35]; its assumptions (base entropy convergence, an O(Delta x^delta) correction, and L-infinity/BV/time-Lipschitz bounds) do not contain the target conclusion, so invoking it is independent support rather than circularity. Lemma 5.1 directly proves the required O(Delta x^delta) bound from the explicit cap (5.4); the cap is a sufficient condition, not a fitted parameter. The numerical EOA uses a fine-mesh solution of the same scheme as a reference, but that is a self-convergence diagnostic, not one of the analytical claims. Finally, the gap between the analyzed capped slopes (5.4) and the implemented slopes (3.2) (Remark 5 requires K to grow as Delta x tends to 0, while Theorem 5.1(ii) needs a fixed K) is a correctness limitation in the theorem's applicability, not a circular reduction of the conclusion to an input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the hypotheses H1-H4, the CFL condition (4.12), and the cited Theorem 5.1 from the authors' prior work. No new physical or mathematical entities are introduced; the only invented ingredients are algorithmic choices (slope limiters, quadrature rules) and the ad-hoc constants K and delta in the slope modification.

free parameters (3)
  • alpha (numerical viscosity coefficient) = 0.16 in experiments; theory requires alpha in (0, 8/27)
    Appears in the Lax-Friedrichs flux (3.13) and in the CFL condition (4.12); its value is chosen by the user and is not fitted to data, but all estimates depend on it.
  • theta (slope limiter angle) = not reported in experiments; theory requires theta in [0, 0.5]
    Controls the minmod limiter in (3.2) and the convolution slope in (3.11); the convergence proof takes theta as a fixed parameter.
  • K and delta (mesh-dependent slope modification) = unspecified; K > 0, delta in (0,1)
    Introduced in (5.4) solely to make the entropy-convergence proof work. Lemma 5.1 needs a uniform K; Remark 5 suggests choosing K depending on the mesh, which is not rigorous in the limit.
assumptions (5)
  • domain assumption Theorem 5.1 (entropy-consistency perturbation criterion) from [35]
    Used in Section 5 to reduce entropy convergence to three conditions; the proof is not reproduced in this paper, only cited.
  • domain assumption First-order Lax-Friedrichs scheme (3.15) converges to the entropy solution of (2.1), per [6,3,1]
    Required by Theorem 5.1(i); the paper cites prior work and does not re-derive it.
  • domain assumption Entropy solutions of (2.1) are unique, per [6,7]
    Used in Definition 2.2 and in the statement of Theorem 5.2; uniqueness makes the convergent subsequence limit the unique solution.
  • standard math Kolmogorov compactness theorem
    Used in Remark 6 and in the derivation of weak convergence from the L-infinity, BV, and time-continuity estimates.
  • domain assumption Hypotheses H1-H4 on the flux f and kernel mu
    Smoothness and growth assumptions used throughout the estimates; they define the class of problems considered.

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Pith. "Pith review of A MUSCL-Hancock scheme for non-local conservation laws." pith.science (2026). https://pith.science/paper/CL2MEWPA

@misc{pith2026250604176,
  author       = {Pith},
  title        = {Pith review of: A MUSCL-Hancock scheme for non-local conservation laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CL2MEWPA}},
  note         = {Machine review of arXiv:2506.04176}
}
abstract

In this article, we propose a MUSCL-Hancock-type second-order scheme for the discretization of a general class of non-local conservation laws and present its convergence analysis. The main difficulty in designing a MUSCL-Hancock-type scheme for non-local equations lies in the discretization of the convolution term, which we carefully formulate to ensure second-order accuracy and facilitate rigorous convergence analysis. We derive several essential estimates including $\mathrm{L}^\infty,$ bounded variation ($\mathrm{BV}$) and $\mathrm{L}^1$- Lipschitz continuity in time, which together with the Kolmogorov's compactness theorem yield the convergence of the approximate solutions to a weak solution. Further, by incorporating a mesh-dependent modification in the slope limiter, we establish convergence to the entropy solution. Numerical experiments are provided to validate the theoretical results and to demonstrate the improved accuracy of the proposed scheme over its first-order counterpart.

Figures

Figures reproduced from arXiv: 2506.04176 by the authors.

Figure 1
Figure 1. Example 1. Log–log plots of L 1 error versus CPU time for the MH and RK-2 schemes applied to (2.1) with the smooth initial condition (6.3) at time t = 0.15. Results are displayed for three different choices of the interval [a, b] in (6.2): (a) [0.0, 0.25] (upstream convolution), (b) [−0.125, 0.125] (centered convolution), and (c) [−0.25, 0.0] (downstream convolution). 3 2 1 0 1 2 3 x 0.0 0.2 0.4 0.6 0.8 1.0 0 (a) 3 … view at source ↗
Figure 2
Figure 2. Example 2. (a) Initial datum given in (6.5). (b) Numerical solutions at time t = 2.5, computed with the kernel function (6.2) with [a, b] = [−0.25, 0.0] using ∆x = 6/150 and ∆t = ∆x/20. and [a, b] = [−0.25, 0.0] (downstream convolution). The numerical solutions are computed in the domain [−3.0, 3.0] up to time t = 2.5 using absorbing boundary conditions and the results are displayed in [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 3
Figure 3. Example 3. (a) Initial datum given in (6.8). (b) Numerical solutions at time t = 0.5, computed with the kernel function (6.7), using ∆x = 3/150 and ∆t = ∆x/20. 10 1 10 2 10 3 CPU time [s] 10 1 L 1 e r r o r RK-2 MH (a) 10 0 10 1 10 2 CPU time [s] 10 2 10 1 L 1 e r r o r RK-2 MH (b) [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Examples 2 and 3. Log–log plots of L 1 error versus CPU time for the MH and RK-2 schemes for discontinuous solutions. (a) Example 2 and (b) Example 3. size ∆x = 3/900. Additionally, we present the L 1 error versus CPU time plots in [PITH_FULL_IMAGE:figures/full_fig_p0…

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Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [35]

    G. D. VEERAPPAGOWDA, K. SUDARSHANKUMAR,ANDN. MANOJ,Convergence of a second-order scheme for non-local conservation laws, ESAIM Math. Model. Numer. Anal., 57 (2023), pp. 3439–3481

    work page 2023

  2. [1]

    AGGARWAL, R

    A. AGGARWAL, R. M. COLOMBO,ANDP. GOATIN,Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), pp. 963–983

    work page 2015

  3. [2]

    AGGARWAL, H

    A. AGGARWAL, H. HOLDEN,ANDG. VAIDYA,On the accuracy of the finite volume approximations to nonlocal conservation laws, Numer. Math., 156 (2024), pp. 237–271

    work page 2024

  4. [3]

    AMORIM, R

    P. AMORIM, R. M. COLOMBO,ANDA. TEIXEIRA,On the numerical integration of scalar nonlocal conservation laws, ESAIM Math. Model. Numer. Anal., 49 (2015), pp. 19–37. 27 APREPRINT- SEPTEMBER5, 2025

    work page 2015

  5. [4]

    BAYEN, J.-M

    A. BAYEN, J.-M. CORON, N. DENITTI, A. KEIMER,ANDL. PFLUG,Boundary controllability and asymptotic stabilization of a nonlocal traffic flow model, Vietnam J. Math., 49 (2021), pp. 957–985

    work page 2021

  6. [5]

    BERTHON,Why the MUSCL-Hancock scheme isL 1-stable, Numer

    C. BERTHON,Why the MUSCL-Hancock scheme isL 1-stable, Numer. Math., 104 (2006), pp. 27–46

    work page 2006

  7. [6]

    BETANCOURT, R

    F. BETANCOURT, R. BÜRGER, K. H. KARLSEN,ANDE. M. TORY,On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), pp. 855–885

    work page 2011

  8. [7]

    BLANDIN ANDP

    S. BLANDIN ANDP. GOATIN,Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), pp. 217–241

    work page 2016

Show all 39 references
  1. [8]

    BÜRGER, P

    R. BÜRGER, P. GOATIN, D. INZUNZA,ANDL. M. VILLADA,A non-local pedestrian flow model accounting for anisotropic interactions and domain boundaries, Math. Biosci. Eng., 17 (2020), pp. 5883–5906

    2020

  2. [9]

    CHALONS, P

    C. CHALONS, P. GOATIN,ANDL. M. VILLADA,High-order numerical schemes for one-dimensional nonlocal conservation laws, SIAM J. Sci. Comput., 40 (2018), pp. A288–A305

    2018

  3. [10]

    CHANDRASHEKAR, B

    P. CHANDRASHEKAR, B. NKONGA, A. K. MEENA,ANDA. BHOLE,A path conservative finite volume method for a shear shallow water model, J. Comput. Phys., 413 (2020), pp. 109457, 29

    2020

  4. [11]

    F. A. CHIARELLO ANDP. GOATIN,Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM Math. Model. Numer. Anal., 52 (2018), pp. 163–180

    2018

  5. [12]

    Networks and Heterogeneous Media, to appear, Aug

    ,Non-local multi-class traffic flow models. Networks and Heterogeneous Media, to appear, Aug. 2018

    2018

  6. [13]

    F. A. CHIARELLO ANDA. TOSIN,Macroscopic limits of non-local kinetic descriptions of vehicular traffic, Kinet. Relat. Models, 16 (2023), pp. 540–564

    2023

  7. [14]

    CIOTIR, R

    I. CIOTIR, R. FAYAD, N. FORCADEL,ANDA. TONNOIR,A non-local macroscopic model for traffic flow, ESAIM Math. Model. Numer. Anal., 55 (2021), pp. 689–711

    2021

  8. [15]

    R. M. COLOMBO, M. GARAVELLO,ANDM. LÉCUREUX-MERCIER,A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), pp. Paper No. 1150023, 34

    2012

  9. [16]

    R. M. COLOMBO, M. HERTY,ANDM. MERCIER,Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 353–379

    2011

  10. [17]

    R. M. COLOMBO ANDM. LÉCUREUX-MERCIER,Nonlocal crowd dynamics models for several populations, Acta Math. Sci. Ser. B (Engl. Ed.), 32 (2012), pp. 177–196

    2012

  11. [18]

    FRIEDRICH ANDO

    J. FRIEDRICH ANDO. KOLB,Maximum principle satisfying CWENO schemes for nonlocal conservation laws, SIAM J. Sci. Comput., 41 (2019), pp. A973–A988

    2019

  12. [19]

    FRIEDRICH, O

    J. FRIEDRICH, O. KOLB,ANDS. GÖTTLICH,A Godunov type scheme for a class of LWR traffic flow models with non-local flux, Netw. Heterog. Media, 13 (2018), pp. 531–547

    2018

  13. [20]

    FRIEDRICH, S

    J. FRIEDRICH, S. SUDHA,ANDS. RATHAN,Numerical schemes for a class of nonlocal conservation laws: a general approach, Netw. Heterog. Media, 18 (2023), pp. 1335–1354

    2023

  14. [21]

    GODLEWSKI ANDP.-A

    E. GODLEWSKI ANDP.-A. RAVIART,Hyperbolic systems of conservation laws, vol. 3/4 of Mathématiques & Applications (Paris) [Mathematics and Applications], Ellipses, Paris, 1991

    1991

  15. [22]

    GOTTLIEB ANDC.-W

    S. GOTTLIEB ANDC.-W. SHU,Total variation diminishing runge-kutta schemes, Math. Comput., 67 (1998), pp. 73–85

    1998

  16. [23]

    GUINOT ANDC

    V. GUINOT ANDC. DELENNE,MUSCL schemes for the shallow water sensitivity equations with passive scalar transport, Comput. & Fluids, 59 (2012), pp. 11–30

    2012

  17. [24]

    KEIMER ANDL

    A. KEIMER ANDL. PFLUG,Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), pp. 4023–4069

    2017

  18. [25]

    ,On approximation of local conservation laws by nonlocal conservation laws, J. Math. Anal. Appl., 475 (2019), pp. 1927–1955

    2019

  19. [26]

    KEIMER, L

    A. KEIMER, L. PFLUG,ANDM. SPINOLA,Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping, J. Math. Anal. Appl., 466 (2018), pp. 18–55

    2018

  20. [27]

    A. Y. LEROUX,Convergence of an accurate scheme for first order quasilinear equations, RAIRO Anal. Numér., 15 (1981), pp. 151–170

    1981

  21. [28]

    MANOJ, G

    N. MANOJ, G. V. GOWDA,ANDS. K. KENETTINKARA,A positivity preserving second-order scheme for multi-dimensional system of non-local conservation laws, arXiv preprint arXiv:2412.18475v2, (2024)

    2024 arXiv

  22. [29]

    SHU ANDS

    C.-W. SHU ANDS. OSHER,Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471. 28 APREPRINT- SEPTEMBER5, 2025

    1988

  23. [30]

    SOPASAKIS ANDM

    A. SOPASAKIS ANDM. A. KATSOULAKIS,Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), pp. 921–944

    2006

  24. [31]

    W. TONG, R. YAN,ANDG. CHEN,On a class of robust bound-preserving MUSCL-Hancock schemes, J. Comput. Phys., 474 (2023), pp. Paper No. 111805, 21

    2023

  25. [32]

    ,A class of bound-preserving MUSCL-Hancock schemes in two dimensions, J. Comput. Phys., 498 (2024), pp. Paper No. 112668, 28

    2024

  26. [33]

    B.VANLEER,Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov’s method, J. Comput. Phys., 32 (1979), pp. 101–136

    1979

  27. [34]

    ,On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Statist. Comput., 5 (1984), pp. 1–20

    1984

  28. [36]

    VIALLON,Convergence of the two-point upstream weighting scheme, Math

    M.-C. VIALLON,Convergence of the two-point upstream weighting scheme, Math. Comp., 57 (1991), pp. 569– 584

    1991

  29. [37]

    VILA,High-order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws, Math

    J.-P. VILA,High-order schemes and entropy condition for nonlinear hyperbolic systems of conservation laws, Math. Comp., 50 (1988), pp. 53–73

    1988

  30. [38]

    ,An analysis of a class of second-order accurate Godunov-type schemes, SIAM J. Numer. Anal., 26 (1989), pp. 830–853

    1989

  31. [39]

    WAAGAN,A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics, J

    K. WAAGAN,A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics, J. Comput. Phys., 228 (2009), pp. 8609–8626. 29

    2009

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