Convergence of fully discrete Godunov and Lax–Friedrichs schemes for general $n\times n$ systems of hyperbolic conservation laws

For a strictly hyperbolic $n\times n$ system of conservation laws in one space dimension with small total-variation initial data, the unique entropy-admissible solution is generated by a Lipschitz semigroup (the standard Riemann semigroup, equivalently the vanishing-viscosity limit). The convergence analysis of the two oldest fully discrete approximations — the Godunov scheme (1959) and the Lax–Friedrichs scheme — remains, by the explicit statement of Bressan–Chiri–Shen (Arch. Ration. Mech. Anal. 2021) and of Bressan's survey (J. Hyperbolic Differ. Equ. 2024), an open problem: no proof is known that these approximations converge unconditionally to the semigroup solution, and no convergence rates are known, because the total-variation estimates that power the Glimm-scheme and front-tracking theories are provably unavailable (resonance counterexamples of Baiti–Bressan–Jenssen). Subsequent work (Bressan–Guerra 2024; Chiri–Zhang 2024; Giesselmann–Sikstel 2026; Leotta 2025) strengthens the conditional, a-posteriori side of the theory — including a conditional $\varepsilon^{1/3}$ rate and a non-constructive universal convergence modulus under a small-total-variation hypothesis — but leaves the unconditional convergence question for fully discrete schemes on general $n\times n$ systems open. The problem asks for an unconditional convergence proof (with rates) for Godunov and/or Lax–Friedrichs, or an explicit counterexample to convergence or limit identification.

Background

A system of conservation laws $u_t+f(u)_x=0$ in one space dimension, with $u$ taking values in $\mathbb{R}^n$ and flux $f:\mathbb{R}^n\to\mathbb{R}^n$, is the basic model of nonlinear waves in continuum physics (the Euler equations of gas dynamics being the canonical example). The system is called strictly hyperbolic if the Jacobian matrix $Df(u)$ has $n$ real, distinct eigenvalues $\lambda_1(u)<\dots<\lambda_n(u)$ with eigenvectors $r_i(u)$. A characteristic field $i$ is genuinely nonlinear if $\nabla\lambda_i(u)\cdot r_i(u)$ never vanishes (such fields carry shocks and rarefaction waves), and linearly degenerate if $\nabla\lambda_i(u)\cdot r_i(u)\equiv 0$ (such fields carry contact discontinuities). Because smooth solutions develop discontinuities in finite time, one works with weak solutions selected by an admissibility condition; the standard reference class uses functions of bounded variation (BV, meaning finite total variation Tot.Var.), and for initial data $\bar u$ with sufficiently small total variation the Cauchy problem is well posed: there is a unique global solution obtained as a Lipschitz continuous semigroup $S_t\bar u$ (the standard Riemann semigroup, constructed as the limit of Glimm's random-choice scheme or of front-tracking approximations, and equivalent to the zero-viscosity limit of $u^\delta_t+f(u^\delta)_x=\delta u^\delta_{xx}$ by Bianchini–Bressan). This semigroup trajectory is the accepted notion of the exact solution.

The classical fully discrete schemes discretize both time and space. The Godunov scheme evolves piecewise-constant data by solving, at each cell interface, a Riemann problem (the self-similar Cauchy problem with two constant initial states) for one time step and then replacing the solution by its cell averages; the Lax–Friedrichs scheme averages over staggered cells, which introduces numerical viscosity. Both are first-order, conservative, and entropy-dissipating, and are run under a CFL condition (a stability restriction tying $\Delta t$ to $\Delta x$ and the characteristic speeds). They are the foundational schemes of computational gas dynamics, yet their convergence theory for systems is incomplete. In the scalar case $n=1$, monotone schemes converge with rates. For systems, compensated compactness — a compactness technique that derives strong convergence from entropy dissipation via Young measures — yields convergence essentially only for $2\times 2$ systems (DiPerna 1983) and for the isentropic Euler system (Ding–Chen–Luo 1989), and gives no uniqueness and no rates. By contrast, Glimm's random-choice scheme, which advances the solution by sampling the exact Riemann solution at randomly chosen points inside each cell instead of averaging, converges with the sharp rate $\mathcal{O}(\varepsilon^{1/2}|\log\varepsilon|)$ (Glimm 1965; Ancona–Marson 2011); front tracking, which approximates the exact solution by piecewise-constant functions whose elementary waves interact exactly, converges; the semidiscrete upwind scheme (time discretized, space continuous) converges (Bianchini 2003); and vanishing-viscosity approximations converge (Bianchini–Bressan 2005).

The obstruction for fully discrete schemes is twofold. First, the averaging step at each time step creates artificial wave interactions that the Glimm-functional interaction estimates cannot control. Second, and decisively, Baiti–Bressan–Jenssen constructed $2\times 2$ resonant systems for which the total variation of Godunov approximations is amplified by an arbitrarily large factor (when a shock speed is close to a rational multiple of the grid ratio $\Delta x/\Delta t$), and proved the analogous instability for Lax–Friedrichs discrete shock profiles, i.e. traveling-wave profiles of the scheme joining two constant states. Hence no uniform a priori BV bound — the backbone of all known convergence proofs for systems — is available for these schemes.

Bressan–Chiri–Shen (2021) bypassed this obstruction with an a posteriori theory: if the computed approximation dissipates entropy and its total variation verifiably remains below the threshold of the semigroup domain (checked by a post-processing algorithm that also traces the large shocks and measures oscillations on trapezoidal subdomains), then an explicit $L^1$ error bound against the semigroup solution is obtained, applicable to Godunov, Lax–Friedrichs, backward Euler, and mollification approximations. The guarantee is conditional: if the total-variation check fails, the algorithm aborts and no estimate is produced. That paper states verbatim that "a major remaining open problem is the convergence of fully discrete approximations, such as the Lax–Friedrichs or the Godunov scheme", and its remarks predict, heuristically, error sizes of order $\varepsilon^{1/3}$ (or $\varepsilon^{1/3}|\log\varepsilon|$ when shocks are present) in favorable piecewise-Lipschitz regimes — rates not proven unconditionally for numerical outputs.

Problem Statement

Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\times n$ system of conservation laws in one space dimension with $f\in C^2$, each characteristic field genuinely nonlinear or linearly degenerate, and let the initial data $\bar u$ have sufficiently small total variation, so that the Cauchy problem generates the standard Riemann semigroup $S_t\bar u$ (equivalently, the vanishing-viscosity solution). Let $u^\varepsilon$ be the approximate solutions produced by the first-order Godunov scheme or by the Lax–Friedrichs scheme, under the CFL stability condition, with mesh size $\varepsilon=\Delta x\to 0$ and time step $\Delta t=O(\varepsilon)$.

(1) Convergence and limit identification: prove, or disprove by an explicit counterexample, that $u^\varepsilon(t,\cdot)\to S_t\bar u$ in $L^1_{\mathrm{loc}}(\mathbb{R})$, uniformly for $t$ in bounded intervals. A disproof may exhibit a strictly hyperbolic system and small-BV data for which the approximations fail to converge, or converge to a limit that is not the semigroup trajectory.

(2) Convergence rate: in the affirmative case, establish a rate or explicit modulus $\varrho(\varepsilon)\to 0$ such that $\|u^\varepsilon(t,\cdot)-S_t\bar u\|_{L^1}\le\varrho(\varepsilon)$, and compare it with the heuristic $\varepsilon^{1/3}$ and $\varepsilon^{1/3}|\log\varepsilon|$ scales of Bressan–Chiri–Shen and the $\varepsilon^{1/2}|\log\varepsilon|$ conjecture for the universal modulus.

The analysis must be unconditional in the BV sense: no step may assume, as a hypothesis to be verified a posteriori on the computed output (as in the post-processing of Bressan–Chiri–Shen), that $\mathrm{Tot.Var.}\{u^\varepsilon(t,\cdot)\}$ remains uniformly small. A resolution for either scheme (Godunov or Lax–Friedrichs) alone is admissible. The same convergence question for the backward Euler time discretization (time discrete, space continuous), known completely only for scalar laws, is an explicitly attached secondary sub-question from the same source; progress on it alone does not discharge the fully discrete core.

References

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