Convergence of Godunov-compatible finite volume schemes for systems of hyperbolic balance laws: the unverified Godunov-scheme uniqueness/stability and per-scheme compactness hypotheses (Ben-Artzi–Li framework)

Ben-Artzi and Li (Math. Comp. 90, 2021) proved a Lax–Wendroff-type convergence theorem for compact high-order finite volume schemes for one-dimensional systems of nonlinear hyperbolic balance laws: a consistent, conservative scheme whose iterates are uniformly $L^\infty$-bounded and compact in $C([0,T];L^1_{\mathrm{loc}})$ converges to a solution of the balance law, and if the scheme is Godunov compatible the limit is the unique entropy solution — but only under an unproved hypothesis (Assumption 5.4) that the first-order Godunov scheme itself converges to a unique entropy solution with an $L^1$-stability bound, verified so far only for scalar laws, a class of 2×2 systems, and isentropic gas dynamics. The open problem, explicitly flagged by the authors and unresolved in the subsequent literature through 2026, is to complete this theory: establish the Godunov-scheme uniqueness/stability hypothesis for broader classes of hyperbolic systems (or rigorously delimit its validity, given the known grid-resonance instability of Bressan–Jenssen–Baiti), verify the required boundedness/compactness hypotheses for concrete high-order schemes such as MUSCL and GRP in discontinuous regimes, and extend the flux-consistency and Godunov-compatibility framework to nonuniform grids and to fluxes more general than $f(u)$.

Background

Systems of nonlinear hyperbolic balance laws $u_t+f(u)_x=0$ with $u(x,t)\in\mathbb{R}^D$ express conservation of mass, momentum, and energy in compressible flow; their solutions develop discontinuities (shock waves) in finite time even from smooth initial data, so solutions are understood in the weak (distributional) sense. Among weak solutions one selects the physically relevant entropy solution, characterized by a family of entropy inequalities; for general systems, uniqueness of entropy solutions is itself a largely open problem, known in the scalar case (a single conserved quantity, $D=1$) and for restricted systems and data classes.

The Godunov finite volume scheme advances cell averages by solving, at each cell interface, a Riemann problem — the exact initial-value problem with piecewise-constant left and right constant states — and averaging the result back over the cell. High-resolution extensions — the MUSCL schemes of van Leer (MUSCL: Monotonic Upstream-centered Scheme for Conservation Laws, finite volume schemes with piecewise-linear data reconstruction), the GRP (generalized Riemann problem) scheme of Ben-Artzi and Falcovitz, which solves Riemann problems with piecewise-linear initial data, and modern ADER (arbitrary-derivative Riemann solver), WENO (weighted essentially non-oscillatory reconstruction) and DG (discontinuous Galerkin) solvers — replace the piecewise-constant data by piecewise-linear or higher-order reconstructions, so each mesh cell carries several degrees of freedom (cell values, slopes). The classical Lax–Wendroff theorem (Lax and Wendroff, 1960) guarantees only that a consistent, conservative first-order scheme with convergent iterates has a weak-solution limit; it says nothing about the entropy condition and does not directly cover schemes computing several quantities per cell.

Ben-Artzi and Li (Math. Comp. 90, 2021) extended the notion of consistency to such compact high-order finite volume schemes through approximate fluxes and proved a generalized Lax–Wendroff theorem (their Theorem 4.1): if the discrete iterates are uniformly bounded in $L^\infty$ and converge in $C([0,T];L^1_{\mathrm{loc}})$, the limit solves the balance law. To conclude that the limit is the unique entropy solution, they introduced Godunov compatibility (Definition 5.5) — the scheme must track the first-order Godunov scheme at discontinuities — and proved (Theorem 5.7, Corollary 5.10) uniqueness and the entropy property of the limit under Assumption 5.4: the Godunov scheme itself converges to a unique solution of the balance law and satisfies the $L^1$-stability estimate $\|\Phi_{k,G}\theta-\Phi_{k,G}\psi\|_1\le(1+Ck)\|\theta-\psi\|_1$ for initial data in a bounded ball. This assumption is documented only for scalar laws, for a class of 2×2 systems (DiPerna, 1983), and for isentropic gas dynamics (Ding–Chen–Luo, 1989); for broader systems it is unproved — and it is known to fail as a uniform statement in general, since Bressan, Jenssen, and Baiti (2006) constructed a 2×2 strictly hyperbolic system (the flux Jacobian $df(u)$ has real, distinct eigenvalues) and a resonant solution for which the Godunov iterates oscillate without bound, showing that no a priori BV bounds or $L^1$-stability estimates can hold for finite-difference schemes in full generality. The authors also deliberately restrict the framework to uniform grids in one space dimension (their Remark 3.6 records that nonuniform grids are avoided because of notational complications) and to fluxes $f(u)$ depending only on $u$, noting that no general framework exists for more general flux functions.

Problem Statement

Complete the convergence theory of Godunov-compatible high-order finite volume schemes for one-dimensional systems of nonlinear hyperbolic balance laws by removing its currently unverified hypotheses, as explicitly left open by Ben-Artzi and Li (Mathematics of Computation 90 (2021), 141–169):

(1) Godunov-scheme hypothesis (Assumption 5.4 of the source). Prove that the first-order Godunov finite volume scheme converges to the unique entropy solution and satisfies the $L^1$-stability estimate for a class of hyperbolic systems strictly larger than the known cases (scalar laws; the 2×2 class treated by DiPerna; isentropic gas dynamics). The target is any class of systems and initial data that is strictly larger than these three known cases (i.e. not contained in any of them) and is defined by explicit, stated conditions. Or, conversely, rigorously delimit the classes of systems and initial data for which the hypothesis fails, going beyond the known grid-resonance instability counterexample of Bressan–Jenssen–Baiti (which already rules out a uniform stability bound in full generality).

(2) Per-scheme stability and compactness (hypotheses (i)–(ii) of Theorem 4.1 of the source). For at least one emblematic family of high-order compact schemes (for example MUSCL- or GRP-type schemes) in a regime with discontinuous solutions, establish the uniform $L^\infty$ bounds and the a priori compactness yielding convergence in $C([0,T];L^1_{\mathrm{loc}})$, or show rigorously that such bounds fail for that scheme.

(3) Framework extensions deferred by the source authors: adapt the flux-consistency definition, the generalized Lax–Wendroff convergence theorem, and Godunov compatibility to nonuniform spatial grids and to fluxes more general than $f(u)$ (depending on additional variables or having discontinuous coefficients), or identify where the framework breaks down. This component requires both extensions: the general-flux case together with the nonuniform-grid adaptation.

These components are independent sub-questions that jointly serve one scientific objective: to turn the conditional statement 'every convergent, consistent, Godunov-compatible high-order finite volume scheme converges to the unique entropy solution' from a theorem under unverified hypotheses into an unconditional theorem for a precisely specified class of systems, schemes, and grids. Each component stands on its own, and no component is a prerequisite for another.

References

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