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

Abstract: 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.