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

Louis Petri, Sigrun Ortleb, Gunnar Birke, Christian Engwer, Hendrik Ranocha

TL;DR

To address stiffness and diffusion-limited asymptotics in linear kinetic models on cut-cell meshes, the paper develops a DoD-stabilized DG method within an SBP framework. The telegraph equation in diffusion scaling with micro-macro decomposition is used as a prototype to prove semidiscrete stability and asymptotic preservation under periodic SBP operators. It analyzes DoD stabilization for central and upwind fluxes and proves that the DoD approach yields stable, AP discretizations that recover the heat equation in the limit $\varepsilon \to 0$. Numerical experiments verify convergence, asymptotic limits, and the viability of implicit time integration in the heat limit.

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.

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

TL;DR

To address stiffness and diffusion-limited asymptotics in linear kinetic models on cut-cell meshes, the paper develops a DoD-stabilized DG method within an SBP framework. The telegraph equation in diffusion scaling with micro-macro decomposition is used as a prototype to prove semidiscrete stability and asymptotic preservation under periodic SBP operators. It analyzes DoD stabilization for central and upwind fluxes and proves that the DoD approach yields stable, AP discretizations that recover the heat equation in the limit . Numerical experiments verify convergence, asymptotic limits, and the viability of implicit time integration in the heat limit.

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.
Paper Structure (25 sections, 7 theorems, 86 equations, 5 figures, 1 table)

This paper contains 25 sections, 7 theorems, 86 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The telegraph equation
  4. Spatial semidiscretization of the base scheme
  5. Semidiscretization in matrix-vector/SBP formulation
  6. Time discretization of the base scheme
  7. The asymptotic limit
  8. Semidiscrete stability and asymptotic analysis
  9. Semidiscrete stability
  10. Formal asymptotic limit
  11. The cut-cell problem and the DoD stabilization
  12. The cut-cell problem
  13. The domain-of-dependence stabilization
  14. DoD stabilization for the telegraph equation
  15. SBP properties of the DoD stabilization
  16. ...and 10 more sections

Key Result

theorem 3.1

Consider the semidiscretization eq:semidiscretization_matrixform of the telegraph equation using a pair of periodic (upwind) SBP operators $\bigl(D^{\varrho}, D^{\tilde{g}}\bigr)$ of the type eq:D1_D2_pairs. Then, this semidiscrete system is stable with respect to the weighted inner-product $\left(u where $\underline{M} = \operatorname{diag}(M, \varepsilon^2 M)$.

Figures (5)

  • Figure 1: One-dimensional cut-cell setting.
  • Figure 2: $L^2$ errors for the DoD-stabilized telegraph equation at the time $T=1$.
  • Figure 3: $L^2$ errors for the DoD-stabilized heat equation at the time $T=1$.
  • Figure 4: Numerical validation of the asymptotic behavior for the DoD-stabilized telegraph equation for $\varepsilon \to 0$ at the time $T=0.5$. The plots show the regularization error measured in the $L^2$-norm for the numerical flux pairs $(D^-,D^+)$ and $(D^z,D^z)$, with polynomial degrees $0,1,2$ and for the two time-stepping methods ARS(4,4,3) and IMEX SSP2(3,2,2).
  • Figure 5: Simulation results for the heat equation, using the implicit midpoint RK scheme at the time $T=5$.

Theorems & Definitions (20)

  • remark 2.1
  • definition 2.2
  • theorem 3.1
  • proof
  • theorem 3.2
  • proof
  • definition 4.1: Extension operator
  • remark 4.2
  • remark 4.3
  • remark 4.4
  • ...and 10 more