arXiv subjects

Sigrun Ortleb

Publications and source records attributed to Sigrun Ortleb.

5 recordsLinked to original sources

arXiv2026

Stable and asymptotic preserving space-time discretizations of a linear kinetic transport equation in diffusive scaling

We develop an unconditionally energy-stable tensor-product space-time discretization framework for the solution of a linear kinetic transport equation in one space dimension. The kinetic equation is a simplified model of radiative transfer formulated as a hyperbolic balance law in diffusive scaling for a particle distribution function of the independent variables space, time and velocity. Our numerical discretization is based on the well-known technique of micro-macro decomposition which results in a system of balance laws for equilibrium and non-equilibrium quantities and facilitates preservation of the asymptotic limit for vanishing scaling parameters at the discrete level. We prove fully discrete stability and asymptotic preservation for general spatial and temporal discretizations having the summation-by-parts property. A new provably energy-stable Dirichlet boundary treatment for the micro-macro decomposed system is developed based on the introduction of simultaneous approximation terms. Numerical results show convergence for smooth problems and demonstrate energy stability of the proposed boundary treatment.

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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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arXiv2025

Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators

The Active Flux method is a numerical method for conservation laws using a combination of cell averages and point values as independent degrees of freedom, based on ideas from finite volumes and finite differences. This unusual mix has been shown to work well in many situations. We expand the theoretical justifications of the Active Flux method by analyzing it from the point of view of summation-by-parts (SBP) operators, which are routinely used to analyze finite difference, finite volume, and finite element schemes. We investigate in what type of setting the Active Flux method can be formulated using classical or degenerate SBP operators, yielding a first and novel approach for showing the energy stability of the Active Flux method. We present the analysis for the one-dimensional scalar linear advection equation with periodic boundary conditions on a uniform grid.

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arXiv2023

On the Stability of IMEX Upwind gSBP Schemes for 1D Linear Advection-Diffusion Equations

A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts~(upwind gSBP) schemes in space and implicit-explicit Runge-Kutta~(IMEX-RK) schemes in time. Hereby, advection terms are discretized explicitly while diffusion terms are solved implicitly. In this context, specific combinations of space and time discretizations enjoy enhanced stability properties. In fact, if the first and second-derivative upwind gSBP operators fulfill a compatibility condition, the allowable time step size is independent of grid refinement, although the advective terms are discretized explicitly. In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin~(DG) schemes on a global level. While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay~(BR1) scheme for diffusion does not allow for grid-independent stable time steps, the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping. Furthermore, unlike previous discrete energy stability investigations for DG schemes, the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space since no finite-element-type inverse constants are involved. Numerical experiments are provided confirming these theoretical findings.

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arXiv2016

Patankar-Type Runge-Kutta Schemes for Linear PDEs

We study the local discretization error of Patankar-type Runge-Kutta methods applied to semi-discrete PDEs. For a known two-stage Patankar-type scheme the local error in PDE sense for linear advection or diffusion is shown to be of the maximal order ${\cal O}(\Delta t^3)$ for sufficiently smooth and positive exact solutions. However, in a test case mimicking a wetting-drying situation as in the context of shallow-water flows, this scheme yields large errors in the drying region. A more realistic approximation is obtained by a modification of the Patankar approach incorporating an explicit testing stage into the implicit trapezoidal rule.

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