Open hypotheses of the Ben-Artzi–Li theory of flux-consistent finite volume schemes: Godunov-scheme convergence and stability for systems of hyperbolic balance laws

Ben-Artzi and Li (Math. Comp. 2021) built a consistency–stability–convergence theory for compact high-order finite volume schemes approximating one-dimensional systems of nonlinear hyperbolic balance laws $u_t + f(u)_x = 0$: a flux-based consistency notion of order $q$, a generalized Lax–Wendroff convergence theorem (their Theorem 4.1), and an entropy-selection mechanism (Godunov compatibility, their Theorem 5.7 and Corollary 5.10) by which a high-order scheme inherits the Godunov scheme's limit. The theory is conditional: Theorem 4.1 assumes uniform $L^\infty$ bounds and convergence in $C([0,T];L^1_{\mathrm{loc}})$ of the approximations, and the entropy conclusions rest on Assumption 5.4 — that the Godunov scheme itself converges to a unique limit satisfying an $L^1$ stability estimate — verified in full only for scalar laws, whereas for a class of $2\times 2$ systems and for isentropic gas dynamics only the paper's relaxed hypothesis (Remark 5.8: all Godunov limit functions are entropy solutions, which does not ensure uniqueness or the stability estimate) is known. The theory is also restricted to uniform one-dimensional grids and autonomous fluxes $f(u)$. No work cited below discharges these hypotheses beyond the scalar case: the open problem is to verify (or refute) the Godunov-scheme uniqueness/stability hypothesis beyond the classes where it or its relaxation is known, to establish the stability/compactness hypotheses for concrete high-order compact schemes, and to extend the flux-consistency and Godunov-compatibility framework to nonuniform grids, several space dimensions, and non-autonomous or discontinuous fluxes, so that the convergence and entropy-selection theorems become unconditional for identified classes of systems and schemes.

Background

Systems of nonlinear hyperbolic conservation laws $u_t + f(u)_x = 0$ with $u(x,t) \in \mathbb{R}^D$ arise throughout continuum physics (gas dynamics, shallow water, traffic flow). Even smooth initial data develop discontinuities (shocks) in finite time, so solutions are understood as weak (distributional) solutions; since weak solutions are not unique, one imposes an entropy condition — a convex entropy $\eta$ with entropy flux $q$ satisfying $\eta(u)_t + q(u)_x \le 0$ in the sense of distributions — to select the physical solution. For systems, existence and uniqueness of entropy solutions are open in general.

Finite volume schemes (FVS) discretize the conservation law in its integral (balance-law) form: cell averages are updated by numerical fluxes through cell interfaces. The Godunov scheme computes the interface flux from the exact solution of a Riemann problem — the initial-value problem with two constant states separated by a jump — averaged over one time step under a CFL condition (a bound on the time-step-to-mesh ratio guaranteeing finite propagation speed). Practical high-resolution codes are compact schemes storing several degrees of freedom per mesh cell (e.g., slopes or moments) and upgrading the Godunov flux: MUSCL (monotone upstream-centered scheme for conservation laws, second-order reconstruction from cell slopes), GRP (generalized Riemann problem, a solver starting from piecewise-linear data that yields analytical second-order fluxes), ADER (arbitrary high-order schemes built from high-order Riemann solvers in time), WENO (weighted essentially non-oscillatory reconstructions, which avoid spurious oscillations near discontinuities), and DG (discontinuous Galerkin) methods, which evolve polynomial data per cell.

The classical Lax–Wendroff theorem (1960) guarantees that if a conservative scheme, consistent in the pointwise sense $g(\xi,\dots,\xi) = f(\xi)$, produces bounded approximations that converge with suitable compactness, then the limit is a weak solution. This theorem does not cover compact high-order schemes with several degrees of freedom per cell, nor does pointwise consistency capture what happens at discontinuities.

Ben-Artzi and Li (2021) rebuilt this foundation in the balance-law framework for one space dimension. They first showed that weak solutions with bounded, continuous cell masses satisfy the integral balance law, the key step being Lipschitz regularity in time of the integrated fluxes $\int_{t_1}^{t_2} f(u(x,t))\,dt$ at cell boundaries. They defined flux-based consistency of order $q$ for approximate fluxes relative to the finite-dimensional approximation space $V^k$ per cell — the same numerical flux can have different orders on different spaces: the Godunov flux is of 'infinite order' on piecewise-constant data but of order zero on piecewise-linear data, while GRP/MUSCL fluxes are first-order consistent for general data and second-order only for smooth data. Their generalized Lax–Wendroff theorem (Theorem 4.1) states that if the FVS is flux-consistent of order $q > 0$ and (i) the approximations are uniformly bounded in $L^\infty$ and (ii) converge in $C([0,T]; L^1_{\mathrm{loc}})$ (or admit a priori compactness yielding such convergence), then the limit is a solution of the balance law. For entropy selection they introduced Godunov compatibility (Definition 5.5): the scheme coincides with the Godunov scheme on piecewise-constant data, and its one-step update stays within $o(k)$ in total $L^1$ of the Godunov update applied to the cell averages, uniformly on admissible data sets. Under Assumption 5.4 — the Godunov FVS applied to piecewise-constant projections of data in a ball $\mathfrak{B}_K$ converges to a unique solution of the balance law, and satisfies the $L^1$ stability estimate $\|\Phi^k_G \theta - \Phi^k_G \psi\|_1 \le (1 + Ck)\|\theta - \psi\|_1$ — Theorem 5.7 shows that all limit functions of a Godunov-compatible scheme coincide with the Godunov limit, and Corollary 5.10 concludes entropy-solution limits for the isentropic Euler system via the relaxed hypothesis of Remark 5.8 — requiring only that all Godunov limit functions be entropy solutions — combined with the convergence theorem of Ding–Chen–Luo (1989) for the Godunov scheme.

Full Assumption 5.4 — a unique limit together with the $L^1$ stability estimate — is known only in the scalar case (where entropy solutions are unique and the Godunov scheme is an $L^1$ contraction). For a class of $2\times 2$ systems, DiPerna's compensated compactness theory (a technique that recovers strong convergence from weak convergence of entropy-dissipation measures, developed in his 1983 paper on convergence of approximate solutions to conservation laws), and for the isentropic Euler system, Ding–Chen–Luo (1989), establish only the relaxed entropy-limit hypothesis: all limits of Godunov approximate solutions are entropy solutions. As the paper's Remark 5.8 states, this requirement is not sufficient to ensure uniqueness, and it does not yield the stability estimate. For general systems, convergence of the Godunov scheme — and even existence/uniqueness of entropy solutions — is a major open problem. An alternative modern route, developed by Fjordholm, Käppeli, Mishra and Tadmor (2017), proves convergence of Godunov-type schemes for arbitrary systems to entropy measure-valued solutions, i.e., limits described by probability measures in phase space (Young measures) rather than by single functions — a strictly weaker conclusion than Assumption 5.4.

Problem Statement

The problem is to complete the Ben-Artzi–Li theory of flux-consistent finite volume approximations to systems of nonlinear hyperbolic balance laws by discharging its unverified hypotheses and removing its declared scope restrictions, so that the convergence and entropy-selection theorems become unconditional for identified classes of systems and schemes. Concretely, for one-dimensional systems $u_t + f(u)_x = 0$ with Riemann-solver-based Godunov fluxes:

1. (Godunov-scheme hypothesis; the deepest open core.) Verify Assumption 5.4 of Ben-Artzi–Li (2021) for a class of hyperbolic systems beyond the scalar case, the only class for which it is known in full — in particular, supplying the uniqueness and stability components for classes such as the DiPerna class of genuinely nonlinear $2\times 2$ systems or the isentropic Euler system covered by Ding–Chen–Luo, where only the relaxed entropy-limit hypothesis is established, would itself be an advance: prove that the Godunov finite volume scheme $\Phi^k_G$ applied to piecewise-constant projections of initial data in the ball $\mathfrak{B}_K$ converges to a unique solution of the balance law and satisfies the $L^1$ stability estimate

$$\|\Phi^k_G \theta - \Phi^k_G \psi\|_1 \le (1 + Ck)\|\theta - \psi\|_1,$$

with $C$ depending only on $K$ (here $k$ is the time step and $\theta$, $\psi$ denote the piecewise-constant cell-average projections of two initial functions $u_0, v_0 \in \mathfrak{B}_K$); or prove that specific parts of this hypothesis fail for an identified class, thereby delimiting the reach of the entropy-selection theorems.

2. (Per-scheme stability hypotheses.) For one or more concrete compact high-order schemes — for example, the GRP scheme or the MUSCL scheme evolving piecewise-linear data — establish the hypotheses (i) uniform $L^\infty$ bounds and (ii) convergence in $C([0,T]; L^1_{\mathrm{loc}})$ (or a priori compactness yielding it) required by Theorem 4.1, beyond the scalar Burgers prototype with artificial viscosity already treated in the literature.

3. (Grid and dimension generality.) Adapt the flux-consistency definition, the generalized Lax–Wendroff convergence theorem, and the Godunov-compatibility notion (with the resulting entropy-selection theorems) to nonuniform spatial grids and to problems in several space dimensions, either of which may be addressed independently.

4. (Flux generality.) Extend the framework to fluxes not of the autonomous form $f(u)$ — fluxes depending on additional variables or on discontinuous coefficients, either of which may be addressed independently — within a single general consistency and convergence framework, replacing the current case-by-case theory.

These four items are the missing hypotheses and the declared scope limitations of one and the same theory; their common scientific objective is that, for the identified classes, every Godunov-compatible compact high-order scheme is rigorously guaranteed to converge to an entropy solution selected by the Godunov limit. A full resolution covers all four items, and resolving any single item is a partial result; a resolution of item 1 alone, for a genuinely broader class of systems, constitutes the deepest open core. For items 3 and 4, the extended framework is to reduce, in the uniform-grid one-dimensional autonomous-flux setting, to the source theory as stated.

References

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