Lukáčová-Medviďová and Yuan (arXiv:2105.02165; Numer. Methods Partial Differ. Equ. 39(5):3777–3810, 2023) proved that the classical first-order Godunov finite volume method, based on the exact Riemann solver, converges for the multidimensional complete compressible Euler equations: the scheme is entropy stable and consistent, the numerical solutions converge weakly to a dissipative measure-valued (DMV) solution, and their Cesàro averages and first variances converge strongly via K-convergence. The entire analysis rests on the physical non-degeneracy Assumption 3.1 — a uniform positive lower bound on the discrete density and a uniform upper bound on the discrete total energy, equivalent to uniform strict convexity of the entropy Hessian — which excludes vacuum and unbounded energy growth. The open problem is to establish this convergence theory without the non-degeneracy assumption, identifying what limit object and what convergence statements survive when vacuum forms or energy becomes unbounded, and to extend the consistency–entropy-dissipation–DMV–K-convergence framework beyond the first-order exact-Riemann-solver setting to general multidimensional hyperbolic systems of conservation laws. In the existing convergence proofs of this program — for relative-energy error estimates, discontinuous Galerkin, thermodynamically compatible, residual-distribution, and flux-corrected finite element schemes, and for a second-order generalized-Riemann-problem scheme for scalar laws — the non-degeneracy-type hypothesis is always assumed; the vacuum core remains open.

Background

The complete (non-isentropic) compressible Euler equations govern an inviscid gas through conservation of mass, momentum, and total energy, closed by a thermodynamic equation of state; for a perfect gas with adiabatic exponent γ\gamma they form, for the conserved variables (ρ,m,E)(\rho,\mathbf{m},E) (density, momentum, total energy) on a periodic multidimensional domain, a hyperbolic system supplemented by the physical entropy inequality for the entropy S=ρcvlog⁡(p/ργ)S=\rho c_v\log(p/\rho^\gamma). In several space dimensions, solutions of hyperbolic conservation laws develop shocks, contact discontinuities, and persistent fine-scale oscillations — vortex sheets in the Kelvin–Helmholtz problem, spike-and-bubble mixing in the Richtmyer–Meshkov problem — and weak entropy solutions are not unique. The question "does the numerical scheme converge, and to what?" therefore has to be posed against a generalized solution concept. The appropriate one here is the dissipative measure-valued (DMV) solution introduced for the complete Euler system by Březina and Feireisl: a Young measure, that is, a family (Vt,x)(\mathcal{V}_{t,x}) of probability distributions of the state variables, which satisfies the conservation laws in expectation, satisfies the entropy inequality, and controls concentration defects; it describes the statistics of all possible weak limits of approximate solution sequences. A companion analytical tool is K\mathcal{K}-convergence, based on Komlós' theorem: even when single approximate solutions only converge weakly and oscillate, their Cesàro averages over a sequence of discretizations converge strongly, so the expected value and the first variance of the limiting Young measure become computable observables.

The Godunov finite volume method is the classical first-order scheme for such systems: the numerical solution is represented by piecewise-constant cell averages, and each time step evolves every cell interface by the exact solution of the local one-dimensional Riemann problem (the initial-value problem with two constant states) — the exact Riemann solver. Lukáčová-Medviďová and Yuan proved for the multidimensional complete Euler system that this scheme is entropy stable and consistent (their Theorem 3.1), that the numerical solutions converge weakly to a DMV solution (Theorem 4.1), that the Cesàro averages and first variances converge strongly by K\mathcal{K}-convergence (Theorem 4.2), and that the numerical solutions themselves converge strongly whenever the limit coincides with a weak or Lipschitz-continuous strong solution (Theorem 4.3). The entire chain, however, rests on the physical non-degeneracy Assumption 3.1: there exist constants ρ‾>0\underline{\rho}>0 and E‾\overline{E} with

0<ρ‾≤ρh,Eh≤E‾0<\underline{\rho}\le\rho_h,\qquad E_h\le\overline{E}

uniformly as the mesh size h→0h\to0 on the time interval [0,T][0,T]. By Lemma B.4 of the same paper this is equivalent to the Hessian of the mathematical entropy being uniformly strictly positive definite in the conserved variables. This uniform strict convexity is what converts entropy dissipation into the weak BV (bounded-variation) estimate and what keeps the wave speeds and the consistency error of the exact Riemann solver under control. Physically, the assumption excludes precisely the delicate regimes: vacuum, where the density degenerates to zero and the entropy Hessian degenerates with it, and unbounded energy growth. The source closes by stating that the numerical analysis of the Godunov method for multidimensional systems "still remains open in general" and names the extension of the results to general multidimensional hyperbolic conservation laws as future work.

Problem Statement

Overarching objective: a rigorous convergence theory for the Godunov finite volume discretization of the multidimensional complete compressible Euler equations that does not presuppose the physical non-degeneracy of the discrete solutions, together with the extension of that theory beyond the first-order exact-Riemann-solver setting. The open problem left by Lukáčová-Medviďová and Yuan consists of the following linked directions:

(1) Core — remove or relax Assumption 3.1. Prove, or disprove, that the first-order Godunov finite volume method (scheme (2.13) of the source), applied to the complete compressible Euler system with perfect-gas equation of state on a periodic multidimensional domain, converges as h→0h\to0 to a dissipative measure-valued solution — with the entropy inequality, the K\mathcal{K}-convergence of Cesàro averages and first variances, and the strong-convergence statements — without the uniform lower bound on density and upper bound on total energy; equivalently, without uniform strict convexity of the entropy Hessian. For the affirmative branch this includes identifying what limit object and what convergence statements survive when vacuum forms (inf⁡ρh→0\inf\rho_h\to0) or the energy grows unbounded (sup⁡Eh→∞\sup E_h\to\infty) and characterizing how the scheme can be controlled in these regimes; a disproof is already complete once it exhibits a configuration in which, without the non-degeneracy assumption, one of the asserted conclusions fails (convergence to a dissipative measure-valued solution, the entropy inequality, or the K\mathcal{K}- and strong-convergence statements), so these descriptive clauses are not a precondition for the negative branch.

(2) Extension explicitly named by the source: extend the full framework — consistency, entropy dissipation, weak convergence to a DMV solution, and K\mathcal{K}-convergence — from the first-order Godunov scheme for the complete Euler system to general multidimensional hyperbolic systems of conservation laws.

Direction (1) for the Euler–Godunov case is the central missing piece: every existing convergence proof in the program imposes the same non-degeneracy-type hypothesis. Direction (2) is the source's explicitly named extension of the same convergence program; partial results exist for higher-order and general-system variants of the framework, but none removes the non-degeneracy hypothesis. Directions (1) and (2) are linked in aim but independent as deliverables: each is a self-contained target, neither requires the other, and a solver need not complete both.

References

  1. Mária Lukáčová-Medviďová and Yuhuan Yuan, Convergence of first-order Finite Volume Method based on Exact Riemann Solver for the Complete Compressible Euler Equations, arXiv:2105.02165 (2021). https://arxiv.org/abs/2105.02165
  2. Mária Lukáčová-Medviďová and Yuhuan Yuan, Convergence of first-order finite volume method based on exact Riemann solver for the complete compressible Euler equations, Numerical Methods for Partial Differential Equations 39(5):3777–3810 (2023), doi:10.1002/num.23025. https://doi.org/10.1002/num.23025
  3. Mária Lukáčová-Medviďová, Bangwei She, and Yuhuan Yuan, Error Estimates of the Godunov Method for the Multidimensional Compressible Euler System, Journal of Scientific Computing 91(3):71 (2022), doi:10.1007/s10915-022-01843-6, arXiv:2111.05009. https://doi.org/10.1007/s10915-022-01843-6
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  12. Mária Lukáčová-Medviďová, Ferdinand Thein, Gerald Warnecke, and Yuhuan Yuan, A note on relations between convexity and concavity of thermodynamic functions, Continuum Mechanics and Thermodynamics 38(5):87 (2026), doi:10.1007/s00161-026-01522-z. https://doi.org/10.1007/s00161-026-01522-z
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