Convergence rates of monotone schemes for conservation laws with discontinuous flux

Adrian Montgomery Ruf; Jayesh Badwaik

REVIEW 2 major objections 4 minor 40 references

Convergence rates of monotone schemes for conservation laws with discontinuous flux

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Pith's one-line read Monotone finite volume schemes for conservation laws with a discontinuous, strictly monotone flux converge at the rate O(√Δx) in L1.

desk verdict First rate proof for nonlinear discontinuous flux is plausible, but Lemma 4.7's trace-regularity passage needs a patch before the claim is fully certified. read the letter →

arxiv 1908.08772 v2 pith:OE4L5PVY submitted 2019-08-23 math.NA cs.NA

Jayesh Badwaik , Adrian Montgomery Ruf This is my paper · ORCID

classification math.NAcs.NA MSC 35L6565M0865M1235R05

keywords scalarconservationlawsdiscontinuousfluxfinitevolumemethodsconvergenceratemonotoneschemesdoubling-of-variablesestimatesinitial-boundaryvalueproblemstemporaltotalvariation

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

This paper proves that monotone upwind finite volume schemes for scalar conservation laws with a flux that is strictly monotone in the conserved variable and a spatial coefficient that is piecewise constant with finitely many jumps converge to the unique entropy solution at the rate $O(\sqrt{\Delta x})$ in $L^1$. If true, this is the first convergence-rate proof for a numerical method in the nonlinear discontinuous-flux setting, and it matches the optimal rate known for conservation laws with a constant-in-space flux. The proof cuts the domain at the discontinuities of the coefficient, turning the problem into finitely many initial-boundary value problems with a flux that no longer depends on $x$, and controls the new interface terms with a discrete flux-matching condition and a temporal total variation bound. If the theorem holds, the error of the scheme is bounded uniformly in the grid size, which provides the quantitative error control needed for adaptive mesh refinement and multilevel Monte Carlo methods.

What carries the argument

The central object is the decomposition of the discontinuous-flux problem into a finite set of initial-boundary value problems on the intervals between the discontinuities of $k$, coupled by the discrete flux-matching condition $u^{n+1}_{P_i} = (f^{(i)})^{-1}\bigl(f^{(i-1)}(u^{n+1}_{P_i-1})\bigr)$ at each interface. The argument is carried by a comparison lemma using the doubling of variables with a mollified test kernel, which estimates the $L^1$ difference between the exact and numerical solutions in terms of moduli of continuity, the grid parameters, and an entropy-dissipation term. The new ingredient that makes the boundary terms manageable is the bound on the temporal total variation of the numerical solution, $\sum_n \lvert u^{n+1}_j - u^n_j \rvert \le C\, \mathrm{TV}(u_0)$, uniform in the cell index; together with the strict monotonicity of the flux this gives the Lipschitz-in-space regularity of the exact solution's flux (Lemma 4.7), which closes the estimate of the interface terms.

What would settle it

Check the temporal total variation of $u(0+,t)$ for the two-flux problem with $g(u) = u$ and $f(u) = u^2/2$ and a Riemann-type initial datum that sends a shock into the interface. If this quantity is unbounded, Lemma 4.7's regularity premise fails and the proof of Theorem 4.10 collapses, so the rate bound is not established.

Watch

Extended reading notes

Core claim

The central claim is that the finite volume approximation defined by (3.1) satisfies $\lVert u(\cdot,T) - u_{\Delta t}(\cdot,T) \rVert_{L^1(\mathbb{R})} \le C \sqrt{\Delta x}$ whenever the flux is strictly monotone, the coefficient $k$ is piecewise constant with finitely many discontinuities, and the initial datum lies in $L^1 \cap BV(\mathbb{R})$. The proof obtains this by decomposing the entropy solution and the numerical solution on the intervals between consecutive discontinuities of $k$, where the restriction of the exact solution solves an initial-boundary value problem with a spatially homogeneous flux. The discretization enforces the flux-matching condition (the Rankine–Hugoniot condition) across each discontinuity on the discrete level, which reduces the global error to a sum of errors on these subdomains. For each subdomain, a standard doubling-of-variables comparison estimate bounds the $L^1$ error, with the interface terms controlled by a new bound on the temporal total variation of the numerical solution (Lemma 4.6) and by a Lipschitz-in-space property of the flux of the exact solution (Lemma 4.7). Together these ingredients give the stated $\sqrt{\Delta x}$ rate on the whole line.

Load-bearing premise

The proof assumes that the boundary trace $u(0+,t)$ of the exact solution on each interface has bounded temporal total variation, so that $w = f(u)$ is Lipschitz in $x$ with values in $L^1(0,T)$; the paper states this follows from the discrete bound of Lemma 4.6 'and carries over in the limit' without demonstrating the limit passage.

Editorial extensions

If this is right

The scheme (3.1) converges to the entropy solution with an error of at most $C\sqrt{\Delta x}$, giving the first convergence-rate statement for nonlinear conservation laws with discontinuous flux.
For each fixed number of discontinuities, the global $L^1$ error on the whole real line obeys the same $\sqrt{\Delta x}$ bound as the error on any single subdomain, so the rate does not degrade when summing across interfaces.
The same proof strategy yields an $O(\sqrt{\Delta x})$ error bound for general initial-boundary value problems whose boundary data lie in $L^1\cap BV(0,T)$, improving on the previously claimed $O(\Delta x^{1/3})$ for that setting.
The $\sqrt{\Delta x}$ rate is optimal without extra assumptions on the initial datum, since the standard optimality construction for the homogeneous case can be placed away from the last discontinuity.

Reading between the lines

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

If the temporal total variation of the interface trace is not uniformly bounded for some admissible data, the rate may degrade or the scheme may fail to reach $\sqrt{\Delta x}$ order; this is a testable gap in the proof rather than a demonstrated counterexample.
A Wasserstein-distance analogue of this rate, analogous to the homogeneous case, may be provable with the same decomposition and temporal-total-variation control; the paper names this as a future direction.
The strict monotonicity assumption on the flux is probably not essential for the decomposition idea; non-monotone fluxes could be split into monotone branches, though the interface coupling would then require a new matching condition.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here's my read of Badwaik–Ruf. The headline result is genuinely new: Theorem 5.1 gives the first O(sqrt(Delta x)) rate for monotone upwind schemes for scalar conservation laws with discontinuous, nonlinear flux, in the strictly monotone, piecewise-constant coefficient case. The proof strategy is sensible: split at flux discontinuities, treat each interval as an IBVP with flux-matched boundary data, glue via discrete Rankine–Hugoniot. The new ingredient is the temporal TV bound on the numerical solution at interface ghost cells (Lemma 4.6), which feeds into boundary terms in the Kuznetsov machinery. That part is detailed and I found no algebra issues.

The paper is not quite complete as written. Two passages are under-verified. The gluing in Section 2 is a sketch, but in this flux-monotone setting it is believable and likely works. More important is Lemma 4.7. The exact trace u(0+,t) is asserted to have bounded temporal TV because the discrete trace does and 'carries over in the limit.' That is not automatic from L1 convergence plus uniform BV_x; the stress-test example (v_n = 1_{[0,1/n]} -> 0) shows why. The paper itself flags the missing support. The good news is that a direct argument exists: on the left half-line, u is the entropy solution and lies in BV(R x (0,T)), so the trace u(0-,t) has bounded time variation; the Rankine–Hugoniot matching f(u(0+,t)) = g(u(0-,t)) transfers this to the right trace. That patch is short and does not disturb the rest. So the gap is real but repairable.

Minor: numerical experiments have no code or data, and the scheme is non-conservative at the interface (the rate implies the defect vanishes). The citation pattern is honest: self-citations supply auxiliary facts, not the target rate.

Net: the central claim is likely true and this is the first rate for this setting once Lemma 4.7 is patched. A serious referee should engage. It belongs in the literature after revision expanding the gluing and trace passages. I'd take it to reading group and would cite the revised version. Send it to peer review.

Referee Report

2 major / 4 minor

Summary. The paper proves an O(sqrt(Delta x)) L1 convergence rate for a class of monotone finite volume schemes approximating scalar conservation laws with discontinuous flux, under the assumptions that the flux is strictly monotone in the unknown and the spatial coefficient is piecewise constant with finitely many discontinuities. The proof decomposes the problem across the flux discontinuities into finitely many initial-boundary value problems, establishes Kuznetsov-type error estimates for the half-line and bounded-interval cases, and assembles these into Theorem 5.1. The appendix extends the technique to general initial-boundary value problems with prescribed boundary data. If correct, this is the first convergence-rate proof for nonlinear schemes in the discontinuous-flux setting.

Significance. The main theorem is significant: for nonlinear conservation laws with discontinuous flux, no convergence-rate result for numerical schemes was previously available, and the decomposition strategy plus the discrete temporal-total-variation bound are potentially reusable tools. The paper also gives a self-contained Kuznetsov framework for half-line problems and extends the rate to general IBVPs, which is of independent interest. The authors are appropriately careful that the rate sqrt(Delta x) is optimal in the absence of extra assumptions, citing the classical Sabac example. However, the proof as written has a load-bearing gap in Lemma 4.7, and one further reduction in Section 2 is asserted rather than shown.

major comments (2)

[Section 4.2, Lemma 4.7] The Lipschitz bound ∫_0^T |f(u(x,t))-f(u(y,t))| dt ≤ C|x-y| is obtained by applying [31, Lem. 4] to w=f(u), and the required temporal total-variation bound of the boundary trace u(0+,t) is said to 'follow from Lemma 4.6 on a discrete level and carries over in the limit.' That limit passage is not demonstrated, and it is not automatic: uniform BV_x plus strong L1 convergence does not imply convergence of one-sided traces, as the sequence v_n = 1_{[0,1/n]} converges to 0 in L1 with BV=1 while the trace at 0+ stays 1. Since the J2 term in Theorem 4.10 is estimated by Cε exactly through Lemma 4.7, and J2 controls the boundary contribution on R+, this gap is load-bearing for Theorem 4.10 and hence for Theorem 5.1. The authors should supply a direct argument for the exact entropy-solution trace (for example, from BV regularity in (x,t) on R− combined with the Rankine–Hugoniot condition) or prove trace convergence of the numerical solutions by an explicit estimate.
[Section 2, composite solution assertion] The paragraph after (2.4) asserts that if u(0) and u(i) are entropy solutions of the respective IBVPs, then the composite u = Σ_i u(i) is the entropy solution of (1.1) 'by adding the entropy inequalities of u(i) and choosing the respective constant in each entropy inequality in accordance with (2.1).' This reduction is used in Theorem 5.1 to obtain the global rate from rates on each subdomain, so the summation must be shown explicitly. In particular, the trace terms at each interface ξ_i involve limits from both sides and must combine correctly under the discrete Rankine–Hugoniot condition; the authors should display this calculation so that no residual interface term remains.

minor comments (4)

[Abstract and title] There are numerous typographical errors, for example 'ra tes' in the abstract and 'conser v a tion' in the title; the manuscript needs a careful proofreading pass.
[Lemma 4.9 proof] The notation φ^n = φ(x,t_n,y,s) and φ_{j+1/2} = φ(x_{j+1/2},t,y,s) is introduced but the arguments of the mollifiers are not always written consistently in the sums; please clarify and check all occurrences.
[Table 1] The observed orders of convergence in both experiments exceed 1 for the finest grids (1.28 and 1.30), which is not predicted by Theorem 5.1; the authors should comment that these are pre-asymptotic effects or otherwise explain the discrepancy.
[Appendix A] The generalization to arbitrary BV boundary data is plausible, but the modified Lemma 4.6 is only described ('should then read') rather than proved, and the dependence of the constants on TV(a) is not tracked; please provide the proof or at least a precise statement with explicit constants.

Circularity Check

0 steps flagged · score 1.0 of 10

No construction-level circularity: the sqrt(Delta x) rate follows from explicit Kuznetsov-type estimates with epsilon = epsilon0 = sqrt(Delta x) chosen by balancing error terms; the main self-citation supplies auxiliary regularity, not the target rate. One flagged regularity gap in Lemma 4.7 is a proof gap, not a circular reduction.

full rationale

The central derivation is self-contained. In Theorem 4.4 and Theorem 4.10 the convergence rate is obtained by proving explicit estimates of the form C(Delta x + Delta t + epsilon + epsilon0 + Delta x/epsilon + Delta x/epsilon0 + Delta t/epsilon0) and then choosing epsilon = epsilon0 = sqrt(Delta x), a standard balance of error terms rather than a fitted parameter renamed as a prediction. The numerical method and the entropy solution are not defined in terms of the claimed rate. The Rankine-Hugoniot decomposition in Section 2 and the discrete Rankine-Hugoniot interface condition in (3.1) are used as logical links, not as circular assumptions. The self-citations are auxiliary: [31] supplies a regularity lemma and a TVD fact for bounded domains, [33] supplies an optimality result in the Wasserstein distance that is not used in the proof of the main rate, and [34] is an external optimality example. The proof does not reduce to its conclusion by construction. The flagged limitation is in Lemma 4.7: the proof that f(u) is Lipschitz in space with values in L1(0,T) invokes [31, Lem.4] and states that the required temporal total variation bound of the trace u(0+,t) 'follows from Lemma 4.6 on a discrete level and caries over in the limit.' This passage from the discrete numerical trace to the exact trace is not demonstrated, and Theorem 4.10's J2 estimate depends on Lemma 4.7. This is a genuine proof gap that belongs under correctness risk, not circularity: Lemma 4.6 is proved independently for the numerical approximation, and the imported Lipschitz regularity is not the convergence rate being claimed. Therefore the circularity score is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data: the only tunable quantities are the mollifier widths eps and eps0, chosen as sqrt(Delta x) in the final optimization, and the mesh ratio lambda, all standard analysis parameters. The argument relies on the cited well-posedness and trace theory and on two regularity statements that are only sketched in the text.

assumptions (6)

domain assumption Well-posedness, uniqueness and trace existence for entropy solutions of (1.1) with adapted entropies.
Invoked in Remark 2.3 via Baiti-Jensen [6] and Andreianov et al. [3]; the numerical comparison is against this unique entropy solution.
domain assumption The flux is strictly monotone in u, f_u >= alpha > 0.
Section 2 assumption; makes each f^(i) invertible and gives the boundary update formula (3.1) a Lipschitz inverse.
domain assumption Initial datum u0 in (L1 cap BV)(R).
Used for total variation bounds, modulus of continuity estimates, and the Crandall-Majda discrete entropy argument.
domain assumption CFL condition (3.2) holds.
Needed for monotonicity, TVD and the discrete entropy inequality in Lemmas 4.2, 4.8 and 4.9.
standard math Standard Kuznetsov estimates for spatially independent conservation laws from [8, 15] apply.
Used in Lemmas 4.2 and 4.8 to bound the A and B terms on the half line.
domain assumption The regularity assertion in Lemma 4.7 that w = f(u) is Lipschitz in space with values in L1(0,T).
The proof invokes [15, Thm 2.15] and [31, Lem 4] and asserts the temporal total variation of u(0+,t) carries over from the discrete level to the limit; that passage is not shown in detail.

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Pith. "Pith review of Convergence rates of monotone schemes for conservation laws with discontinuous flux." pith.science (2026). https://pith.science/paper/OE4L5PVY

@misc{pith2026190808772,
  author       = {Pith},
  title        = {Pith review of: Convergence rates of monotone schemes for conservation laws with discontinuous flux},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OE4L5PVY}},
  note         = {Machine review of arXiv:1908.08772}
}

abstract

We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of $\sqrt{\Delta x}$ in $\mathrm{L}^1$, whenever the flux is strictly monotone in $u$ and the spatial dependency of the flux is piecewise constant with finitely many discontinuities. We also present numerical experiments to illustrate the main result. To the best of our knowledge, this is the first proof of any type of convergence rate for numerical methods for conservation laws with discontinuous, nonlinear flux. Our proof relies on convergence rates for conservation laws with initial and boundary value data. Since those are not readily available in the literature we establish convergence rates in that case en passant in the Appendix.

Figures

Figures reproduced from arXiv: 1908.08772 by the authors.

**Figure 2.** Numerical solution of Experiment 2 with ∆ [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Mishra, and G

Adimurthi, S. Mishra, and G. V. Gowda , Conservation law with the ﬂux function dis- continuous in the space variable—II: Convex–concave type ﬂ uxes and generalized entropy solutions, Journal of Computational and Applied Mathematics, 203 (20 07), pp. 310 – 344

work page

[2] [2]

Misra, and G

Adimurthi, S. Misra, and G. V. Gowda , Optimal entropy solutions for conservation laws with discontinuous ﬂux-functions , Journal of Hyperbolic Diﬀerential Equations, 2 (2005), pp. 783–837

work page 2005

[3] [3]

Andreianov, K

B. Andreianov, K. H. Karlsen, and N. H. Risebro , A theory of L1-dissipative solvers for scalar conservation laws with discontinuous ﬂux , Archive for rational mechanics and analysis, 201 (2011), pp. 27–86

work page 2011

[4] [4]

Audusse and B

E. Audusse and B. Perthame , Uniqueness for scalar conservation laws with discontinuou s ﬂux via adapted entropies , Proceedings of the Royal Society of Edinburgh Section A: Math- ematics, 135 (2005), pp. 253–265

work page 2005

[5] [5]

Multilevel Monte Carlo Finite Volume Methods for Random Conservation Laws with Discontinuous Flux

J. Badw aik, N. H. Risebro, and C. Klingenberg , Multilevel Monte Carlo ﬁnite volume meth- ods for random conservation laws with discontinuous ﬂux , arXiv preprint arXiv:1906.08991, (2019). CONSERVATION LAWS WITH DISCONTINUOUS FLUX 23

work page Pith review arXiv 2019

[6] [6]

Baiti and H

P. Baiti and H. K. Jenssen , Well-posedness for a class of 2 × 2 conservation laws with L∞ data, Journal of Diﬀerential Equations, 140 (1997), pp. 161 – 185

work page 1997

[7] [7]

B ¨urger, K

R. B ¨urger, K. Karlsen, C. Klingenberg, and N. Risebro , A front tracking approach to a model of continuous sedimentation in ideal clariﬁer–thick ener units , Nonlinear Analysis: Real World Applications, 4 (2003), pp. 457–481

work page 2003

[8] [8]

Coclite, J

G. Coclite, J. Ridder, and N. Risebro , A convergent ﬁnite diﬀerence scheme for the Ostrovsky-Hunter equation on a bounded domain , BIT Numerical Mathematics, 57 (2017), pp. 93–122

work page 2017