Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2510.23231

Stability analysis of discontinuous Galerkin with a high order embedded boundary treatment for linear hyperbolic equations

Mirco Ciallella
View original source ↗Read source document ↗Export metadata ↓

Abstract

Embedded, or immersed, approaches have the goal of reducing to the minimum the computational costs associated with the generation of body-fitted meshes by only employing fixed, possibly Cartesian, meshes over which complex boundaries can move freely. However, this boundary treatment introduces a geometrical error of the order of the mesh size that, if not treated properly, can spoil the global accuracy of a high order discretization, herein based on discontinuous Galerkin. The shifted boundary polynomial correction was proposed as a simplified version of the shifted boundary method, which is an embedded boundary treatment based on Taylor expansions to deal with unfitted boundaries. It is used to accordingly correct the boundary conditions imposed on a non-meshed boundary to compensate the aforementioned geometrical error, and reach high order accuracy. In this paper, the stability analysis of discontinuous Galerkin methods coupled with the shifted boundary polynomial correction is conducted in depth for the linear advection equation, by visualizing the eigenvalue spectrum of the high order discretized operators. The analysis considers a simplified one-dimensional setting by varying the degree of the polynomials and the distance between the real boundary and the closest mesh interface. The main result of the analysis shows that the considered high order embedded boundary treatment introduces a limitation to the stability region of high order discontinuous Galerkin methods with explicit time integration, which becomes more and more important when using higher order methods. The implicit time integration is also studied, showing that the implicit treatment of the boundary condition allows one to overcome such limitation and achieve an unconditionally stable high order embedded boundary treatment.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mirco Ciallella. 2025-10-27. Stability analysis of discontinuous Galerkin with a high order embedded boundary treatment for linear hyperbolic equations. https://arxiv.org/abs/2510.23231

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗