Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1606.03032

A strategy to implement Dirichlet boundary conditions in the context of ADER finite volume schemes. One-dimensional conservation laws

Gino I. Montecinos
View original source ↗Read source document ↗Export metadata ↓

Abstract

ADER schemes are numerical methods, which can reach an arbitrary order of accuracy in both space and time. They are based on a reconstruction procedure and the solution of generalized Riemann problems. However, for general boundary conditions, in particular of Dirichlet type, a lack of accuracy might occur if a suitable treatment of boundaries conditions is not properly carried out. In this work the treatment of Dirichlet boundary conditions for conservation laws in the context of ADER schemes, is concerned. The solution of generalized Riemann problems at the extremes of the computational domain, provides the correct influence of boundaries. The reconstruction procedure, for data near to the boundaries, demands for information outside the computational domain, which is carried out in terms of ghost cells, which are provided by using the numerical solution of auxiliary problems. These auxiliary problems are hyperbolic and they are constructed from the conservation laws and the information at boundaries, which may be partially or totally known in terms of prescribed functions. The evolution of these problems, unlike to the usual manner, is done in space rather than in time due to that these problems are named here, {\it reverse problems}. The methodology can be considered as a numerical counterpart of the inverse Lax-Wendroff procedure for filling ghost cells. However, the use of Taylor series expansions, as well as, Lax-Wendroff procedure, are avoided. For the scalar case is shown that the present procedure preserve the accuracy of the scheme which is reinforced with some numerical results. Expected orders of accuracy for solving conservation laws by using the proposed strategy at boundaries, are obtained up to fifth-order in both space and time.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gino I. Montecinos. 2016-06-09. A strategy to implement Dirichlet boundary conditions in the context of ADER finite volume schemes. One-dimensional conservation laws. https://arxiv.org/abs/1606.03032

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↗