High-Order Discontinuous Cut Finite Element Methods for Scalar Hyperbolic Conservation Laws
Abstract
In this paper, we present a family of high-order cut finite element methods based on the discontinuous Galerkin (DG) framework for scalar hyperbolic conservation laws on complex domains. Building on our previous work, we develop a multidimensional formulation that combines macro-element stabilization with flux limiting to obtain a scheme that preserves the maximum principle and remains robust with respect to arbitrary boundary cuts of the background mesh. The physical domain is embedded in a regular background mesh, which may produce arbitrarily small cut cells. To avoid the severe time step restrictions typically associated with such cells, ghost penalty stabilization terms are added on interior facets of macro-elements. The resulting method exhibits stability and accuracy properties similar to those of standard DG methods on fitted meshes. An L2-stability result is derived for the semi-discrete scheme under both periodic and inflow-outflow boundary conditions. To enforce the maximum principle and suppress nonphysical oscillations, we adapt limiter techniques from standard DG methods to the CutFEM setting by defining limiting parameters on macro-elements. In particular, we present a macro-element-based parameterized flux limiter together with adaptations of the Zhang-Shu bound-preserving limiter and the Barth-Jespersen slope limiter. Numerical experiments in two and three spatial dimensions demonstrate optimal convergence orders, preservation of the maximum principle, and accurate shock capturing without spurious oscillations, even for challenging cut configurations involving very small-cut cell intersections.
Related
- Fully discrete multi-entropy-stable compact Runge--Kutta discontinuous Galerkin methods for hyperbolic conservation laws
- Least-Squares Finite Element Methods with Weakly Imposed Boundary Conditions: Norm and Error Equivalences, Over-Penalization, and Fully Discontinuous Discretizations
- 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
- Anisotropic error estimates for the L2-projection onto discontinuous finite element spaces
BibTeX
@misc{fu2026highorder,
title = {{High-Order Discontinuous Cut Finite Element Methods for Scalar Hyperbolic Conservation Laws}},
author = {Pei Fu and Gunilla Kreiss and Zelin Xin and Sara Zahedi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08880},
eprint = {2609.08880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08880},
}
RIS
TY - UNPB
TI - High-Order Discontinuous Cut Finite Element Methods for Scalar Hyperbolic Conservation Laws
AU - Pei Fu
AU - Gunilla Kreiss
AU - Zelin Xin
AU - Sara Zahedi
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08880
UR - https://arxiv.org/abs/2609.08880
AB - In this paper, we present a family of high-order cut finite element methods based on the discontinuous Galerkin (DG) framework for scalar hyperbolic conservation laws on complex domains. Building on our previous work, we develop a multidimensional formulation that combines macro-element stabilization with flux limiting to obtain a scheme that preserves the maximum principle and remains robust with respect to arbitrary boundary cuts of the background mesh. The physical domain is embedded in a regular background mesh, which may produce arbitrarily small cut cells. To avoid the severe time step restrictions typically associated with such cells, ghost penalty stabilization terms are added on interior facets of macro-elements. The resulting method exhibits stability and accuracy properties similar to those of standard DG methods on fitted meshes. An L^2-stability result is derived for the semi-discrete scheme under both periodic and inflow-outflow boundary conditions. To enforce the maximum principle and suppress nonphysical oscillations, we adapt limiter techniques from standard DG methods to the CutFEM setting by defining limiting parameters on macro-elements. In particular, we present a macro-element-based parameterized flux limiter together with adaptations of the Zhang-Shu bound-preserving limiter and the Barth-Jespersen slope limiter. Numerical experiments in two and three spatial dimensions demonstrate optimal convergence orders, preservation of the maximum principle, and accurate shock capturing without spurious oscillations, even for challenging cut configurations involving very small-cut cell intersections.
ER -