arXiv subjects

Gunnar Birke

Publications and source records attributed to Gunnar Birke.

8 recordsLinked to original sources

arXiv2026

The domain-of-dependence stabilization for cut-cell meshes is fully discretely stable

We present a fully discrete stability analysis of the domain-of-dependence stabilization for hyperbolic problems. The method aims to address issues caused by small cut cells by redistributing mass around the neighborhood of a small cut cell at a semi-discrete level. Our analysis is conducted for the linear advection model problem in one spatial dimension. We demonstrate that fully discrete stability can be achieved under a time step restriction that does not depend on the arbitrarily small cells, using an operator norm estimate. Additionally, this analysis offers a detailed understanding of the stability mechanism and highlights some challenges associated with higher-order polynomials. We also propose a way to mitigate these issues to derive a feasible CFL-like condition. The analytical findings, as well as the proposed solution are verified numerically in one- and two-dimensional simulations.

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arXiv2026

On the consistency of the Domain of Dependence cut cell stabilization

So called cartesian cut cell meshes provide efficient ways to generate meshes but do require tailored numerical methods to not suffer from stabilization issues, especially in the hyperbolic regime where the application of explicit time stepping schemes is common. In this scenario, due to potentially arbitrarily small cut cells, an infeasible restriction is imposed on the time step size. The Domain of Dependence (DoD) stabilization allows for a time step size based on the underlying Cartesian mesh. Being an extension of a discontinuous Galerkin (DG) method, one would expect similar accuracy properties as in the pure DG case. While numerical results do support this expectation, on the analytical level this has only been investigated thoroughly for $k=0$. Error analysis typically hinges on a consistency result. In this contribution we prove such a result for the DoD stabilization given an arbitrary polynomial degree and an exact solution of sufficient regularity. This in turn could open the way towards a more refined analysis of the method even in the high-order case.

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arXiv2026

An Energy-Preserving Domain of Dependence Stabilization for the Linear Wave Equation on Cut-Cell Meshes

We present an energy-preserving (either energy-conservative or energy-dissipative) domain of dependence stabilization method for the linear wave equation on cut-cell meshes. Our scheme is based on a standard discontinuous Galerkin discretization in space and an explicit (strong stability preserving) Runge Kutta method in time. Tailored stabilization terms allow for selecting the time step length based on the size of the background cells rather than the small cut cells by propagating information across small cut cells. The stabilization terms preserve the energy stability or energy conservation property of the underlying discontinuous Galerkin space discretization. Numerical results display the high accuracy and stability properties of our scheme.

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arXiv2026

Domain-of-dependence-stabilized cut-cell discretizations of linear kinetic models with summation-by-parts properties

We employ the summation-by-parts (SBP) framework to extend the recent domain-of-dependence (DoD) stabilization for cut cells to linear kinetic models in diffusion scaling. Numerical methods for these models are challenged by increased stiffness for small scaling parameters and the necessity of asymptotics preservation regarding a parabolic limit equation. As a prototype model, we consider the telegraph equation in one spatial dimension subject to periodic boundary conditions with an asymptotic limit given by the linear heat equation. We provide a general semidiscrete stability result for this model when spatially discretized by arbitrary periodic (upwind) SBP operators and formally prove that the fully discrete scheme is asymptotic preserving. Moreover, we prove that DoD with central numerical fluxes leads to periodic SBP operators. Furthermore, we show that adapting the upwind DoD scheme yields periodic upwind SBP operators. Consequently, the DoD stabilization possesses the desired properties considered in the first part of this work and thus leads to a stable and asymptotic preserving scheme for the telegraph equation. We back our theoretical results with numerical simulations and demonstrate the applicability of this cut-cell stabilization for implicit time integration in the heat equation limit.

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arXiv2024

Error analysis of a first-order DoD cut cell method for 2D unsteady advection

In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the $L^\infty$-in-time $L^2$-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results.

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arXiv2023

Domain of Dependence stabilization for the acoustic wave equation on 2D cut-cell meshes

Cut-cell meshes are an attractive alternative to avoid common mesh generation problems. For hyperbolic problems they pose additional challenges, as elements can become arbitrarily small, leading to prohibitive time step restrictions for explicit time stepping methods. To alleviate this small cell problem we consider a particular stabilization method, the Domain of Dependence (DoD) method. So far, while posessing many favorable theoretical properties, in two dimensions the DoD method was essentially restricted to the transport equation. In this work we extend the DoD method to the acoustic wave equation in two dimensions and provide numerical results for validation.

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arXiv2023

DoD stabilization for higher-order advection in two dimensions

When solving time-dependent hyperbolic conservation laws on cut cell meshes one has to overcome the small cell problem: standard explicit time stepping is not stable on small cut cells if the time step is chosen with respect to larger background cells. The domain of dependence (DoD) stabilization is designed to solve this problem in a discontinuous Galerkin framework. It adds a penalty term to the space discretization that restores proper domains of dependency. In this contribution we introduce the DoD stabilization for solving the advection equation in 2d with higher order. We show an $L^2$ stability result for the stabilized semi-discrete scheme for arbitrary polynomial degrees $p$ and provide numerical results for convergence tests indicating orders of $p+1$ in the $L^1$ norm and between $p+\frac 1 2$ and $p+1$ in the $L^{\infty}$ norm.

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arXiv2023

DoD Stabilization of linear hyperbolic PDEs on general cut-cell meshes

Standard numerical methods for hyperbolic PDEs require for stability a CFL-condition which implies that the time step size depends on the size of the elements of the mesh. On cut-cell meshes, elements can become arbitrarily small and thus the time step size cannot take the size of small cut-cells into account but has to be chosen based on the background mesh elements. A remedy for this is the so called DoD (domain of dependence) stabilization for which several favorable theoretical and numerical properties have been shown in one and two space dimensions. Up to now the method is restricted to stabilization of cut-cells with exactly one inflow and one outflow face, i.e. triangular cut-cells with a no-flow face. We extend the DoD stabilization to cut-cells with multiple in- and out-flow faces by properly considering the flow distribution inside the cut-cell. We further prove L2-stability for the semi-discrete formulation in space and present numerical results to validate the proposed extension.

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