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An Energy Stable High-Order Cut Cell Discontinuous Galerkin Method with State Redistribution for Wave Propagation

Christina G. Taylor, Lucas C. Wilcox, Jesse Chan

TL;DR

The paper develops a provably energy-stable, high-order discontinuous Galerkin method for 2D cut meshes by combining a skew-symmetric DG formulation with state redistribution to address the small cell problem in embedded boundaries. It proves that state redistribution acts as a contractive filter in the $L_2$ sense, preserving $L_2$ stability when applied to an energy-stable DG scheme, and demonstrates this through a formal proof and spectral analysis. Numerical experiments on manufactured solutions, the Pacman benchmark, and multi-object fish simulations confirm high-order convergence, energy stability, and a relaxation of the CFL condition due to SRD. The work also provides practical implementation details for embedded boundary representation, merge neighborhoods, and volume quadrature on cut elements, enabling robust wave-propagation simulations in geometrically complex domains with sharp features.

Abstract

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to "cut" a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is $L_2$ energy stable for arbitrary quadrature. We prove that state redistribution can be added to a provably $L_2$ energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's $L_2$ stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.

An Energy Stable High-Order Cut Cell Discontinuous Galerkin Method with State Redistribution for Wave Propagation

TL;DR

The paper develops a provably energy-stable, high-order discontinuous Galerkin method for 2D cut meshes by combining a skew-symmetric DG formulation with state redistribution to address the small cell problem in embedded boundaries. It proves that state redistribution acts as a contractive filter in the sense, preserving stability when applied to an energy-stable DG scheme, and demonstrates this through a formal proof and spectral analysis. Numerical experiments on manufactured solutions, the Pacman benchmark, and multi-object fish simulations confirm high-order convergence, energy stability, and a relaxation of the CFL condition due to SRD. The work also provides practical implementation details for embedded boundary representation, merge neighborhoods, and volume quadrature on cut elements, enabling robust wave-propagation simulations in geometrically complex domains with sharp features.

Abstract

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to "cut" a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is energy stable for arbitrary quadrature. We prove that state redistribution can be added to a provably energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.
Paper Structure (21 sections, 3 theorems, 47 equations, 14 figures, 2 tables)

This paper contains 21 sections, 3 theorems, 47 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods for Addressing the Small Cell Problem
  3. Methods
  4. The DG Formulation
  5. State Redistribution
  6. Proof of $L_2$ Stability of High-Order State Redistribution
  7. Numerical Experiments
  8. Manufactured Solution
  9. Eigenvalues of RHS Operator
  10. Pacman Benchmark
  11. Fish Simulation
  12. Conclusions
  13. Acknowledgments
  14. Appendix: Implementation Details
  15. Embedded Boundary Representation
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $S$ be the state redistribution operator and $u \in L_2(\Omega)$. Then $S$ is contractive,

Figures (14)

  • Figure 1: An example of an embedding Cartesian domain, $\hat{\Omega}$, with embedded boundaries defining regions $\Omega_{E}$ to exclude and the resulting simulation domain $\Omega$. Once meshed, elements remain Cartesian (white), become cut cells (active portion purple and yellow), or get excluded from the simulation domain (dark grey). Note cut meshes can result in extremely small or skewed elements, such as those shown in yellow.
  • Figure 2: The three-step process of state redistribution in 1D over two-small elements and one full-sized element.
  • Figure 3: Pressure field for the manufactured solution at $t=0$ showing the embedded object.
  • Figure 4: $L_2$ error for the manufactured solution at various grid sizes, $h$, and polynomial degree, $N$, at time $t=1.3$.
  • Figure 5: The mesh and embedded object for the eigenvalue experiments, with one of smallest cells highlighted.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof