Fully discrete multi-entropy-stable compact Runge--Kutta discontinuous Galerkin methods for hyperbolic conservation laws
Abstract
We develop fully discrete multi-entropy-stable high-order compact Runge--Kutta discontinuous Galerkin (cRKDG) methods for hyperbolic conservation laws. Our main result is a weak cell entropy inequality for multi-dimensional systems after one full high-order Runge--Kutta (RK) time step. The proof exploits the cell-local intermediate stages of cRKDG: their perturbations are controlled by the initial in-cell variation. The analysis does not require a strong stability preserving decomposition or stagewise entropy correction. For a sufficiently dissipative Lax--Friedrichs flux and explicit RK methods with nonnegative output weights, the inequality holds under a suitable time-step restriction for strongly convex entropy pairs. For a prescribed, uniformly conditioned family of such entropies, common dissipation and time-step conditions ensure weak cell entropy stability for all entropy pairs. A single cell-average-preserving scaling at the end of each step then enforces all corresponding strong cell entropy inequalities simultaneously. We also prove a Lax--Wendroff-type theorem: under the stated assumptions, any strongly convergent sequence of numerical solutions converges to a weak solution satisfying all prescribed entropy inequalities. Numerical experiments in one and two dimensions demonstrate high-order accuracy, the benefit of multiple entropy constraints for nonconvex scalar problems, and robust performance on shock problems.
Related
- Entropy bounded compact stencil Runge-Kutta discontinuous Galerkin method for hyperbolic conservation laws
- A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws
- A merged-cell discontinuous Galerkin method for scalar conservation laws on curved domains
- Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form
- An Interface Green's Function Framework for Complete Discrete W1,∞ Analysis of Discontinuous Galerkin Methods
BibTeX
@misc{sun2026fully,
title = {{Fully discrete multi-entropy-stable compact Runge--Kutta discontinuous Galerkin methods for hyperbolic conservation laws}},
author = {Zheng Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18837},
eprint = {2610.18837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18837},
}
RIS
TY - UNPB
TI - Fully discrete multi-entropy-stable compact Runge--Kutta discontinuous Galerkin methods for hyperbolic conservation laws
AU - Zheng Sun
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18837
UR - https://arxiv.org/abs/2610.18837
AB - We develop fully discrete multi-entropy-stable high-order compact Runge--Kutta discontinuous Galerkin (cRKDG) methods for hyperbolic conservation laws. Our main result is a weak cell entropy inequality for multi-dimensional systems after one full high-order Runge--Kutta (RK) time step. The proof exploits the cell-local intermediate stages of cRKDG: their perturbations are controlled by the initial in-cell variation. The analysis does not require a strong stability preserving decomposition or stagewise entropy correction. For a sufficiently dissipative Lax--Friedrichs flux and explicit RK methods with nonnegative output weights, the inequality holds under a suitable time-step restriction for strongly convex entropy pairs. For a prescribed, uniformly conditioned family of such entropies, common dissipation and time-step conditions ensure weak cell entropy stability for all entropy pairs. A single cell-average-preserving scaling at the end of each step then enforces all corresponding strong cell entropy inequalities simultaneously. We also prove a Lax--Wendroff-type theorem: under the stated assumptions, any strongly convergent sequence of numerical solutions converges to a weak solution satisfying all prescribed entropy inequalities. Numerical experiments in one and two dimensions demonstrate high-order accuracy, the benefit of multiple entropy constraints for nonconvex scalar problems, and robust performance on shock problems.
ER -