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On the consistency of the Domain of Dependence cut cell stabilization

Gunnar Birke, Christian Engwer, Jan Giesselmann, Sandra May

Abstract

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.

On the consistency of the Domain of Dependence cut cell stabilization

Abstract

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 . 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.
Paper Structure (7 sections, 6 theorems, 38 equations)

This paper contains 7 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Base discretization
  4. Extension operators
  5. Consistency of the Domain of Dependence stabilization
  6. Linear advection equation
  7. Linear wave equation

Key Result

lemma 1

Let $X$ and $T$ be vector spaces, $U, W \subset X$ subspaces, $f: U \to T$ and $g: W \to T$ be linear maps. There is a unique linear map such that $(f + g)_{|U} = f$ and $(f + g)_{|W} = g$ if and only if $f_{|U \cap W} = g_{|U \cap W}$, that is the two maps $f$ and $g$ agree on the intersection of $U$ and $W$.

Theorems & Definitions (16)

  • definition 1: Mirroring operator
  • definition 2: Extension operator
  • definition 3: Reflected extension operator
  • lemma 1
  • lemma 2
  • lemma 3
  • proof
  • corollary 1
  • proof
  • definition 4
  • ...and 6 more