A merged-cell discontinuous Galerkin method for scalar conservation laws on curved domains
Abstract
We analyse a discontinuous Galerkin method for two-dimensional scalar conservation laws on curved domains. Small intersections of a Cartesian grid with the domain are combined by rectangular cell merging. The local space is the restriction of ℚk to each physical merged cell. An entropy-conservative flux is supplemented by face viscosity and, for odd degrees, a skew derivative-jump correction on unmerged interior squares. We prove exact semidiscrete L2 dissipation and an optimal error estimate for every fixed k≥1. For smooth solutions with homogeneous trace on the entire boundary, the semi-discrete method satisfies an O(hk+1) error bound. The constants are independent of the original cut-cell area fractions. The proof combines a physical-cell L2 projection and cancellation of regular-face projection errors. A separate semidiscrete result covers linear transport with upwind inflow data. Computations for Burgers' equation on a disk and linear advection on a smooth star exhibit orders close to k+1 for k=1,2,3.
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BibTeX
@misc{liu2026mergedcell,
title = {{A merged-cell discontinuous Galerkin method for scalar conservation laws on curved domains}},
author = {Yong Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32107},
eprint = {2609.32107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32107},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A merged-cell discontinuous Galerkin method for scalar conservation laws on curved domains
AU - Yong Liu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32107
UR - https://arxiv.org/abs/2609.32107
AB - We analyse a discontinuous Galerkin method for two-dimensional scalar conservation laws on curved domains. Small intersections of a Cartesian grid with the domain are combined by rectangular cell merging. The local space is the restriction of ℚ_k to each physical merged cell. An entropy-conservative flux is supplemented by face viscosity and, for odd degrees, a skew derivative-jump correction on unmerged interior squares. We prove exact semidiscrete L^2 dissipation and an optimal error estimate for every fixed k≥1. For smooth solutions with homogeneous trace on the entire boundary, the semi-discrete method satisfies an O(h^k+1) error bound. The constants are independent of the original cut-cell area fractions. The proof combines a physical-cell L^2 projection and cancellation of regular-face projection errors. A separate semidiscrete result covers linear transport with upwind inflow data. Computations for Burgers' equation on a disk and linear advection on a smooth star exhibit orders close to k+1 for k=1,2,3.
ER -