Intrinsic $L^\infty$ bounds and convergence of entropy stable shock-capturing space–time discontinuous Galerkin schemes for multi-dimensional systems of conservation laws

Entropy stable shock-capturing space–time discontinuous Galerkin (SC-DG) methods approximate general nonlinear systems of conservation laws in several space dimensions using entropy variables and entropy stable numerical fluxes. For general systems, their rigorous convergence theory (Hiltebrand and Mishra, 2014; Zakerzadeh and May, 2016) establishes convergence to entropy measure-valued solutions only under an assumed uniform $L^\infty$ bound on the approximate entropy variables — a hypothesis with no known intrinsic justification for multi-dimensional nonlinear systems. The problem is to eliminate this hypothesis: either prove intrinsic uniform $L^\infty$ bounds for the scheme, or otherwise establish mesh-convergence to an entropy measure-valued solution without assuming boundedness. Subsequent developments — the entropy measure-valued framework, statistical solutions, Euler-specific dissipative measure-valued convergence, and 2024–2026 error-estimate work — still rely on assumed boundedness or change the scheme class and solution concept.

Background

A system of hyperbolic conservation laws $\partial_t u + \nabla\!\cdot f(u) = 0$, with $u:\mathbb{R}^d\times\mathbb{R}^+\to\mathbb{R}^N$ and flux $f=(f_1,\dots,f_d)$, models transport-dominated continuum physics (e.g., the compressible Euler equations). Classical solutions develop discontinuities (shocks) in finite time, so one works with weak solutions, and an entropy pair — a convex function $\eta(u)$ (the mathematical entropy, e.g., the negative of physical entropy) together with a companion entropy flux $q(u)$ satisfying $\nabla_u q_j = \nabla_u \eta\, f_j'$ — selects the physically admissible solution via the inequality $\partial_t \eta(u) + \nabla\!\cdot q(u) \le 0$. When the entropy is strictly convex one can change variables to the entropy variables $v = \nabla_u \eta(u)$, in which the system becomes symmetric hyperbolic (symmetrizable); this is the natural variable set for Galerkin discretizations going back to Hughes' entropy-variable finite element formulations. Tadmor's entropy stability theory says a semi-discrete scheme is entropy stable if it satisfies a discrete version of the entropy inequality; it is built from entropy conservative fluxes plus suitable numerical diffusion.

The method at issue (Hiltebrand and Mishra, Numer. Math. 2014) is a fully implicit space–time discontinuous Galerkin (DG) method — DG is a finite element method with discontinuous, elementwise polynomial basis functions, here posed on space–time slabs in the entropy variables — stabilized by streamline diffusion (a consistent linear upwinding term) and a nonlinear residual-based shock-capturing artificial-viscosity term, with entropy stable fluxes at element interfaces. For general nonlinear systems in several space dimensions the authors prove entropy stability and convergence, as the mesh is refined, to an entropy measure-valued (emv) solution: a Young measure, i.e., a map $(x,t)\mapsto \nu_{x,t}$ assigning to each point a probability measure on state space (encoding possible unresolved oscillations), that satisfies the conservation law and the entropy inequality in the sense of averages against $\nu$. Convergence to an actual entropy weak solution is proved only for scalar laws and linear symmetrizable systems. The convergence proof is modeled on the classical approach of DiPerna (Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal. 88 (1985) 223–270): entropy stability gives weak stability of the approximations, but the Young-measure compactness argument requires, as a hypothesis, a uniform bound $\|v^h\|_{L^\infty} \le C$ on the approximate entropy variables, with $C$ independent of the mesh $h$. For scalar laws such bounds are automatic (discrete maximum principles, monotone schemes), but for systems in several space dimensions no mechanism — continuous or discrete — is known that produces them: entropy stability alone yields essentially only $L^2$/energy-type bounds, and the entropy-variable map $u\mapsto v$ can amplify loss of pointwise control. The companion issues the source paper flagged (the shock-capturing scaling parameter $\alpha$, shock-capturing/streamline-diffusion coefficients, and the direct sparse linear solves) concern accuracy and efficiency rather than the convergence theory; the efficient-preconditioning question was subsequently addressed by the authors themselves (Commun. Comput. Phys. 17 (2015) 1320–1359), while the $L^\infty$-bound hypothesis has remained the load-bearing open assumption of the whole framework.

Problem Statement

For the entropy-variable shock-capturing space–time DG scheme of Hiltebrand–Mishra and the streamline-diffusion-free shock-capturing space–time DG variant of Zakerzadeh–May, whose convergence theorems both assume the same mesh-uniform $L^\infty$ bound on the approximate entropy variables, applied to a general nonlinear symmetrizable system of conservation laws $\partial_t u + \nabla\!\cdot f(u)=0$ in $d\ge 2$ space dimensions with initial data $u_0\in L^1\cap L^\infty$ and a strictly convex entropy, eliminate the assumed uniform $L^\infty$ hypothesis for this scheme class: either (a) prove an intrinsic uniform $L^\infty$ bound $\|v^h\|_{L^\infty}\le C$ with $C$ depending only on the data, as a consequence of the schemes' entropy stability and shock-capturing mechanism, and thereby close the convergence-to-emv-solution proof without assumed boundedness; or (b) prove mesh-convergence to an entropy measure-valued solution by any other route that does not presuppose uniform $L^\infty$ bounds. The question concerns the general-systems case; results valid only for scalar equations, linear symmetrizable systems, or a single specific system under additional structural a priori bounds (such as the energy-based dissipative measure-valued framework for Euler) do not by themselves answer it.

References

1. Andreas Hiltebrand and Siddhartha Mishra, "Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws", Numerische Mathematik 126 (2014) 103–151. DOI: 10.1007/s00211-013-0558-0, https://doi.org/10.1007/s00211-013-0558-0 (source paper)
2. Ronald J. DiPerna, "Measure-valued solutions to conservation laws", Archive for Rational Mechanics and Analysis 88 (1985) 223–270. DOI: 10.1007/bf00752112, https://doi.org/10.1007/bf00752112
3. Mohammad Zakerzadeh and Georg May, "On the Convergence of a Shock Capturing Discontinuous Galerkin Method for Nonlinear Hyperbolic Systems of Conservation Laws", SIAM Journal on Numerical Analysis 54 (2016) 874–898. DOI: 10.1137/14096503X, https://doi.org/10.1137/14096503X (preprint arXiv:1404.6030, https://arxiv.org/abs/1404.6030)
4. Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, and Eitan Tadmor, "Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws", Foundations of Computational Mathematics 17 (2017) 763–827. DOI: 10.1007/s10208-015-9299-z, https://doi.org/10.1007/s10208-015-9299-z (preprint arXiv:1402.0909, https://arxiv.org/abs/1402.0909)
5. Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, "On the computation of measure-valued solutions", Acta Numerica 25 (2016) 567–679. DOI: 10.1017/S0962492916000088, https://doi.org/10.1017/S0962492916000088
6. U. S. Fjordholm, S. Lanthaler, and S. Mishra, "Statistical Solutions of Hyperbolic Conservation Laws: Foundations", Archive for Rational Mechanics and Analysis 226 (2017) 809–849. DOI: 10.1007/s00205-017-1145-9, https://doi.org/10.1007/s00205-017-1145-9
7. Eduard Feireisl, Mária Lukáčová-Medviďová, and Hana Mizerová, "Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions", Foundations of Computational Mathematics 20 (2020) 923–966. DOI: 10.1007/s10208-019-09433-z, https://doi.org/10.1007/s10208-019-09433-z (preprint arXiv:1803.08401, https://arxiv.org/abs/1803.08401)
8. 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 (2023) 3777–3810. DOI: 10.1002/num.23025, https://doi.org/10.1002/num.23025
9. Michael Dumbser, Mária Lukáčová-Medviďová, and Andrea Thomann, "Convergence of a hyperbolic thermodynamically compatible finite volume scheme for the Euler equations", Numerische Mathematik 158 (2026) 715–747. DOI: 10.1007/s00211-025-01522-2, https://doi.org/10.1007/s00211-025-01522-2
10. Neelabja Chatterjee and Ulrik Skre Fjordholm, "Convergence of second-order, entropy stable methods for multi-dimensional conservation laws", ESAIM: Mathematical Modelling and Numerical Analysis 54 (2020) 1415–1428. DOI: 10.1051/m2an/2019090, https://doi.org/10.1051/m2an/2019090
11. Hamed Zakerzadeh and Ulrik S. Fjordholm, "High-order accurate, fully discrete entropy stable schemes for scalar conservation laws", IMA Journal of Numerical Analysis 36 (2016) 633–654. DOI: 10.1093/imanum/drv020, https://doi.org/10.1093/imanum/drv020
12. Tianheng Chen and Chi-Wang Shu, "Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws", Journal of Computational Physics 345 (2017) 427–461. DOI: 10.1016/j.jcp.2017.05.025, https://doi.org/10.1016/j.jcp.2017.05.025
13. Xu-Kun Chen, Yong Liu, and Chi-Wang Shu, "The error estimate of entropy-stable discontinuous Galerkin methods for hyperbolic conservation laws" (2026, preprint). arXiv:2608.06642, https://arxiv.org/abs/2608.06642
14. Manting Peng, Zheng Sun, and Kailiang Wu, "OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws", Mathematics of Computation 94 (2025), no. 353, 1147–1198. DOI: 10.1090/mcom/3998, https://doi.org/10.1090/mcom/3998
15. Andreas Hiltebrand and Siddhartha Mishra, "Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws", Communications in Computational Physics 17 (2015) 1320–1359. DOI: 10.4208/cicp.140214.271114a, https://doi.org/10.4208/cicp.140214.271114a
16. Elisabetta Chiodaroli and Ondřej Kreml, "Non-uniqueness of admissible weak solutions to the Riemann problem for isentropic Euler equations", Nonlinearity 31 (2018) 1441–1460. DOI: 10.1088/1361-6544/aaa10d, https://doi.org/10.1088/1361-6544/aaa10d