Convergence of the fully discrete Godunov scheme for general systems of hyperbolic conservation laws

The Godunov upwind scheme is the foundational finite-volume method for systems of conservation laws, yet for general strictly hyperbolic systems no proof is known that its fully discrete approximations converge to the entropy weak solution. Bressan, Jenssen and Baiti (2006) constructed a triangular $2\times2$ system for which grid resonance between a slowly drifting shock and the mesh drives the total variation and the $L^1$ Lipschitz constant of the Godunov approximations to infinity, closing the classical BV/Helly compactness route. Whether the scheme nevertheless converges (and in what sense) for general systems remains open: existing convergence proofs cover only scalar laws, Temple/straight-line systems, and the $2\times2$ isentropic Euler system, and the 2021-2025 conditional a posteriori/a priori error-estimate frameworks assume away exactly the BV instability in question. The problem asks for a convergence theory (or a rigorous non-convergence example) for fully discrete Godunov-type schemes for general systems.

Background

A system of conservation laws in one space dimension is a first-order quasilinear PDE $u_t + f(u)_x = 0$, where $u$ takes values in $\mathbb{R}^n$ and the flux $f$ is smooth; it is strictly hyperbolic when the Jacobian $Df(u)$ has $n$ distinct real eigenvalues (the characteristic speeds). Because solutions develop shocks in finite time, one works with entropy weak solutions: bounded functions satisfying the PDE in distributional sense together with an inequality $\eta(u)_t + q(u)_x \le 0$ for every convex entropy/entropy-flux pair $(\eta,q)$. Classical well-posedness theory (Glimm's scheme, front tracking, vanishing viscosity) yields existence, uniqueness and $L^1$ continuous dependence for initial data of sufficiently small total variation, where the total variation $\mathrm{TV}(u)$ (the BV norm) measures oscillation strength and drives all compactness arguments through Helly's theorem.

The Godunov scheme, introduced in 1959, is the prototype upwind finite-volume method: on a space-time grid with $\Delta x = \Delta t$, the approximate solution is restarted at each time step as piecewise constant cell averages, evolved exactly by solving a Riemann problem (the initial-value problem with two constant states) at each cell interface, and re-averaged; under a CFL condition (the fastest wave crosses at most one cell per step, a constraint named after Courant, Friedrichs and Lewy) this yields a stable, conservative, entropy-consistent recursion. It underlies essentially all modern shock-capturing codes, yet its convergence theory is strikingly incomplete. For scalar equations ($n=1$) monotone-scheme theory gives convergence. For systems, convergence of Godunov's method is known only in special classes: for $2\times2$ and then $n\times n$ straight-line (Temple-class) systems, whose shock curves are straight lines so that same-family interactions create no new oscillation (LeVeque--Temple 1985; Bressan--Jenssen 2000), and for the $2\times2$ isentropic Euler system via compensated compactness (Ding--Chen--Luo 1989) - a technique that extracts strong convergence of approximations from entropy inequalities and Young-measure control, but which so far works only for specially structured systems. In contrast, the Glimm random-sampling scheme, front tracking, vanishing viscosity, and the semidiscrete upwind scheme (discrete in space, continuous in time; Bianchini 2003) all converge for general systems with small-BV data.

Bressan, Jenssen and Baiti (2006) showed that the fully discrete Godunov scheme behaves fundamentally worse on the a priori level. For a triangular $2\times2$ system $u_t + f(u)_x = 0$, $v_t + \lambda v_x + g(u)_x = 0$ with $0<\lambda<f'(u)<1$, chosen so that a $u$-shock travels for a long time at a speed close (but not equal) to a rational such as $1/2$, the mesh resonance between the drifting shock and the lattice generates downstream oscillations in the $v$-component whose total variation grows without bound; the approximations also lose uniform $L^1$ stability with respect to perturbations of the initial data. A companion construction (Baiti--Bressan--Jenssen 2005) showed that the Lax--Friedrichs scheme likewise admits discrete shock profiles (exact discrete traveling-wave solutions) that depend discontinuously on the wave speed, which is precisely what breaks the traveling-wave decomposition underlying the vanishing-viscosity and semidiscrete analyses.

The counterexample thus rules out, for general systems, any proof of convergence via uniform a priori BV bounds and Helly compactness - the standard route for every other approximation method. The authors explicitly leave open whether the Godunov approximations nevertheless converge to the unique entropy weak solution, conjecturing in fact that strong convergence does hold (even for the triangular systems of their own example) and, oppositely, that the resonance instability itself is generic among discrete schemes. Recent work has responded by developing conditional error estimates - a posteriori bounds by Bressan--Chiri--Shen (2021) and Giesselmann--Sikstel (2025), and conditional a priori estimates by Leotta (2025) - all of which assume the numerical solution stays small in BV, precisely the hypothesis the counterexample destroys in general. As stated by Bressan in 2023, a rigorous convergence analysis of the scheme for general solutions containing shocks is still lacking.

Problem Statement

Consider the Cauchy problem for a strictly hyperbolic $n\times n$ system of conservation laws in one space dimension, $u_t + f(u)_x = 0$, $u(0,\cdot) = \bar{u}$, with initial data of sufficiently small total variation, and let $u^{\varepsilon}$ denote the approximate solutions generated by the fully discrete Godunov (upwind) finite-difference scheme on grids with $\Delta x = \Delta t = \varepsilon$ under the standard CFL condition. Determine whether $u^{\varepsilon}$ converges, as $\varepsilon \to 0$, to the unique entropy weak solution of the Cauchy problem - for instance in $L^1_{\mathrm{loc}}$ at each fixed time, or in whatever weaker sense identifies the limit with the entropy solution - for general strictly hyperbolic systems, beyond the special classes (straight-line/Temple systems and the $2\times2$ isentropic Euler system) where convergence is already known. By the Bressan--Jenssen--Baiti counterexample, no proof can proceed through uniform a priori BV bounds or uniform $L^1$ stability, since these fail in general; the question is resolved either by a convergence proof under the stated hypotheses or by a rigorous non-convergence example. The source authors conjecture that convergence to the entropy solution in fact holds, even for the triangular systems of their own counterexample; a rigorous non-convergence example would resolve the question in the opposite direction.

References

1. Alberto Bressan, Helge Kristian Jenssen, Paolo Baiti, An instability of the Godunov scheme, Communications on Pure and Applied Mathematics 59 (2006), no. 11, 1604-1638, DOI 10.1002/cpa.20141, https://doi.org/10.1002/cpa.20141 (arXiv:math/0502125, https://arxiv.org/abs/math/0502125)
2. Randall J. LeVeque, Blake Temple, Stability of Godunov's method for a class of 2x2 systems of conservation laws, Transactions of the American Mathematical Society 288 (1985), no. 1, 115-123, DOI 10.1090/s0002-9947-1985-0773050-x, https://doi.org/10.1090/s0002-9947-1985-0773050-x
3. Alberto Bressan, Helge Kristian Jenssen, On the convergence of Godunov scheme for nonlinear hyperbolic systems, Chinese Annals of Mathematics, Series B 21 (2000), no. 3, 269-284, DOI 10.1142/S0252959900000303, https://doi.org/10.1142/S0252959900000303
4. Xia Xi Ding, Gui-Qiang Chen, Pei-Zhu Luo, Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Communications in Mathematical Physics 121 (1989), no. 1, 63-84, DOI 10.1007/BF01218624, https://doi.org/10.1007/BF01218624
5. Stefano Bianchini, BV solutions of the semidiscrete upwind scheme, Archive for Rational Mechanics and Analysis 167 (2003), no. 1, 1-81, DOI 10.1007/s00205-002-0237-2, https://doi.org/10.1007/s00205-002-0237-2
6. Paolo Baiti, Alberto Bressan, Helge Kristian Jenssen, Instability of travelling wave profiles for the Lax-Friedrichs scheme, Discrete and Continuous Dynamical Systems 13 (2005), no. 4, 877-899, DOI 10.3934/dcds.2005.13.877, https://doi.org/10.3934/dcds.2005.13.877
7. Alberto Bressan, Maria Teresa Chiri, Wen Shen, A posteriori error estimates for numerical solutions to hyperbolic conservation laws, Archive for Rational Mechanics and Analysis 241 (2021), no. 1, 357-402, DOI 10.1007/s00205-021-01653-4, https://doi.org/10.1007/s00205-021-01653-4 (arXiv:2010.00428, https://arxiv.org/abs/2010.00428)
8. Alberto Bressan, One dimensional hyperbolic conservation laws: past and future, arXiv:2310.16707 (2023), https://arxiv.org/abs/2310.16707 (published version Journal of Hyperbolic Differential Equations 21 (2024), no. 3, 523-561, DOI 10.1142/S0219891624400022, https://doi.org/10.1142/S0219891624400022)
9. Fabio Leotta, Conditional a priori error estimates of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D, arXiv:2506.13221 (2025), https://arxiv.org/abs/2506.13221
10. Jan Giesselmann, Aleksey Sikstel, A-posteriori error estimates for systems of hyperbolic conservation laws via computing negative norms of local residuals, IMA Journal of Numerical Analysis 46 (2026), no. 1, 400-430, DOI 10.1093/imanum/drae111, https://doi.org/10.1093/imanum/drae111
11. Fabio Ancona, Stefano Bianchini, Alberto Bressan, Rinaldo M. Colombo, Khai T. Nguyen, Examples and conjectures on the regularity of solutions to balance laws, Quarterly of Applied Mathematics 81 (2023), no. 3, 433-454, DOI 10.1090/qam/1647, https://doi.org/10.1090/qam/1647