On structure-preserving and pointwise conservative continuous DG schemes for hyperbolic systems
Abstract
We present a new class of structure-preserving semi-discrete continuous-discontinuous Galerkin (CG-DG) finite element schemes for linear and nonlinear hyperbolic systems of partial differential equations on unstructured simplex meshes that automatically satisfy the following properties: i) the new schemes are not only cellwise conservative, but also locally pointwise conservative everywhere, hence they satisfy the integral form of the conservation law on arbitrary control volumes that do not have to coincide with the mesh at all; ii) the new methods naturally satisfy the two basic vector calculus identities ∇ · ∇ × A and ∇ × ∇ Z exactly pointwise locally and globally everywhere on the discrete level; iii) for linear symmetric hyperbolic systems the schemes are naturally energy conservative for the square energy, i.e. nonlinearly stable in the L2 norm. The key ingredient of the new CG-DG schemes is the use of two different but compatible approximation spaces: the classical DG space 𝒰hN of discontinuous piecewise polynomials of degree up to N and a classical finite element space 𝒲hN+1 of globally continuous piecewise polynomials of degree N+1. In the new CG-DG schemes, the discrete solution uh is sought in 𝒰hN, while a suitable discrete flux field f̃h is computed in 𝒲hN+1. For N=0 our new schemes are directly related to cell-centered finite volume schemes with suitable vertex-based fluxes. All claimed properties of the schemes are first mathematically proven and are then also verified via suitable numerical tests. We show applications of our approach to three linear and nonlinear hyperbolic systems.
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BibTeX
@misc{abgrall2026structurepreserving,
title = {{On structure-preserving and pointwise conservative continuous DG schemes for hyperbolic systems}},
author = {Rémi Abgrall and Michael Dumbser and Pierre-Henri Maire and Enrico Zampa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.07576},
eprint = {2605.07576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.07576},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On structure-preserving and pointwise conservative continuous DG schemes for hyperbolic systems
AU - Rémi Abgrall
AU - Michael Dumbser
AU - Pierre-Henri Maire
AU - Enrico Zampa
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2605.07576
UR - https://arxiv.org/abs/2605.07576
AB - We present a new class of structure-preserving semi-discrete continuous-discontinuous Galerkin (CG-DG) finite element schemes for linear and nonlinear hyperbolic systems of partial differential equations on unstructured simplex meshes that automatically satisfy the following properties: i) the new schemes are not only cellwise conservative, but also locally pointwise conservative everywhere, hence they satisfy the integral form of the conservation law on arbitrary control volumes that do not have to coincide with the mesh at all; ii) the new methods naturally satisfy the two basic vector calculus identities ∇ · ∇ × A and ∇ × ∇ Z exactly pointwise locally and globally everywhere on the discrete level; iii) for linear symmetric hyperbolic systems the schemes are naturally energy conservative for the square energy, i.e. nonlinearly stable in the L^2 norm. The key ingredient of the new CG-DG schemes is the use of two different but compatible approximation spaces: the classical DG space 𝒰_h^N of discontinuous piecewise polynomials of degree up to N and a classical finite element space 𝒲_h^N+1 of globally continuous piecewise polynomials of degree N+1. In the new CG-DG schemes, the discrete solution u_h is sought in 𝒰_h^N, while a suitable discrete flux field f̃_h is computed in 𝒲_h^N+1. For N=0 our new schemes are directly related to cell-centered finite volume schemes with suitable vertex-based fluxes. All claimed properties of the schemes are first mathematically proven and are then also verified via suitable numerical tests. We show applications of our approach to three linear and nonlinear hyperbolic systems.
ER -