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

Gunnar Birke, Christian Engwer, Jan Giesselmann, Sandra May

Abstract

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.

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

Abstract

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 -in-time -in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results.

Paper Structure

This paper contains 14 sections, 12 theorems, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem setting and main result
  3. Preliminaries
  4. Spatial discretization
  5. Definition of norms
  6. Fully discrete scheme and main result
  7. Geometric cut cell specific estimates
  8. Controlling projection errors
  9. Auxiliary results
  10. Consistency
  11. Discrete dissipation
  12. Boundedness
  13. Proof of main result
  14. Numerical results

Key Result

Lemma 3.1

Let the set of stabilized cut cells $\mathcal{I}$ be given by eq: set of stab cut cells. For a non-stabilized cell $E$, including non-stabilized cut cells, there holds For a stabilized cut cell $E$, there holds with $\alpha_E$ and $\tau$ from eq:newdefalpha For later use of combining the two estimates we define the constant $C_{tr \IfStrEq{X}{X}{}{,X}}:= \max(C_{tr \IfStrEq{1}{X}{}{,1}}, C_{tr

Figures (4)

  • Figure 1: Geometric considerations and mesh construction for ramp test.
  • Figure 2: Geometric considerations for proving projection errors for ramp angle $\gamma < \pi/4$: All non-stabilized cut cells contain triangular cells with the longer leg having a length $\ell \ge \tfrac{h}{2}$. Shown in (a) are examples for four-sided and three-sided cut cells. Figure (b) shows detailed information for this triangle.
  • Figure 3: Convergence results for the time step choice $\Delta t = \frac{1}{5} \frac{h}{ \| \beta \|_{\infty} }$ given in \ref{['eq:theorem cfl']}. Left: Error at time $T$ measured in the $L^2$ (in space) norm. Right: Error at time $T$ measured in the $\beta$-seminorm (in space). The left plot shows convergence orders of $1$ for the error measured in the $L^2$ norm which is better than the order of $0.5$ predicted by theorem \ref{['thm:main']}. The right plot shows convergence orders of $0.5$ for the error measured in the $\beta$-seminorm at the final time $T$ which coincides with the predicted order of theorem \ref{['thm:main']} for the time-averaged $\beta$-seminorm.
  • Figure 4: Convergence results for the time step choice $\Delta t = \frac{1}{2} \frac{h}{ \| \beta \|_{\infty} }$ given in \ref{['eq:usual cfl']}. Left: Error at time $T$ measured in the $L^2$ (in space) norm. Right: Error at time $T$ measured in the $\beta$-seminorm (in space). Also outside of the provable regime we observe the same convergence order of 1 in the $L^2$ norm and of 0.5 in the $\beta$-seminorm, both at the final time $T$, as for the more restrictive time step choice.

Theorems & Definitions (38)

  • Definition 2.1: Space $\mathcal{V}_{*}^0$
  • Definition 2.3
  • Definition 2.5: Upwind and downwind traces
  • Remark 2.6: In/outflow boundary conditions and stabilized cut cells
  • Definition 2.7: $\beta$-weighted mean traces
  • Remark 2.8
  • Definition 2.9: Jump and Average
  • Remark 2.11
  • Definition 2.12: Capacity
  • Definition 2.13: $\beta$-seminorm
  • ...and 28 more