Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Entropy Stable High-Order Discontinuous Galerkin Method on Cut Meshes

Christina G. Taylor, Jesse Chan

TL;DR

The paper develops a high-order entropy-stable discontinuous Galerkin method on Cartesian cut meshes by employing skew-hybridized SBP operators, enabling entropy conservation/stability on arbitrarily cut elements. It constructs polynomially exact, nonnegative-weight quadrature on cut elements through explicit boundary parameterizations, subtriangulation, and Carathéodory pruning, ensuring the SBP property and high-order accuracy. Numerical experiments on the shallow water and compressible Euler equations confirm entropy conservation/stability, $h$-convergence at rate $h^{N+1}$, and robust handling of embedded boundaries, including shocks and complex geometries, with state redistribution offering practical robustness. The approach provides a practical, scalable framework for high-fidelity simulations of hyperbolic systems on cut meshes, with code openly available for reproducibility and further development.

Abstract

High-order entropy stable summation-by-parts (SBP) schemes are a class of robust and accurate numerical methods for hyperbolic conservation laws that are numerically stable at arbitrary order without the need for artificial stabilization. While SBP schemes are well-established on simplicial and tensor-product elements, they have not been extended to cut meshes. Cut meshes provide a convenient and efficient means of mesh generation for domains with embedded boundaries but can be difficult to use due to their arbitrarily shaped cut elements. Using the skew-hybridized SBP formulation of Chan ["Skew-symmetric entropy stable...", JSC, 2019], we present a high-order accurate, entropy stable scheme for hyperbolic conservation laws on cut meshes. The formulation requires positive/non-negative weight quadrature rules on cut elements, which we construct via explicit parameterizations, subtriangulations, and Caratheodory pruning. We numerically verify the accuracy and stability of our method using the shallow water and compressible Euler equations and note promising results for the use of state redistribution with entropy stable methods.

An Entropy Stable High-Order Discontinuous Galerkin Method on Cut Meshes

TL;DR

The paper develops a high-order entropy-stable discontinuous Galerkin method on Cartesian cut meshes by employing skew-hybridized SBP operators, enabling entropy conservation/stability on arbitrarily cut elements. It constructs polynomially exact, nonnegative-weight quadrature on cut elements through explicit boundary parameterizations, subtriangulation, and Carathéodory pruning, ensuring the SBP property and high-order accuracy. Numerical experiments on the shallow water and compressible Euler equations confirm entropy conservation/stability, -convergence at rate , and robust handling of embedded boundaries, including shocks and complex geometries, with state redistribution offering practical robustness. The approach provides a practical, scalable framework for high-fidelity simulations of hyperbolic systems on cut meshes, with code openly available for reproducibility and further development.

Abstract

High-order entropy stable summation-by-parts (SBP) schemes are a class of robust and accurate numerical methods for hyperbolic conservation laws that are numerically stable at arbitrary order without the need for artificial stabilization. While SBP schemes are well-established on simplicial and tensor-product elements, they have not been extended to cut meshes. Cut meshes provide a convenient and efficient means of mesh generation for domains with embedded boundaries but can be difficult to use due to their arbitrarily shaped cut elements. Using the skew-hybridized SBP formulation of Chan ["Skew-symmetric entropy stable...", JSC, 2019], we present a high-order accurate, entropy stable scheme for hyperbolic conservation laws on cut meshes. The formulation requires positive/non-negative weight quadrature rules on cut elements, which we construct via explicit parameterizations, subtriangulations, and Caratheodory pruning. We numerically verify the accuracy and stability of our method using the shallow water and compressible Euler equations and note promising results for the use of state redistribution with entropy stable methods.

Paper Structure

This paper contains 26 sections, 2 theorems, 67 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. High-Order Summation-By-Parts Operators
  3. Constructing Summation-By-Parts Operators on Cut Meshes
  4. Quadrature on Cut Elements
  5. Quadrature Pruning Approaches
  6. Methods
  7. The Skew-Hybridized Summation-by-Parts Formulation
  8. Solution and Test Spaces
  9. Operators
  10. Formulation
  11. Conditions on Quadrature/The Skew-Hybridized SBP Operator
  12. Cut Mesh Generation and Quadrature on Cut Elements
  13. State Redistribution and Entropy Conservation/Stability
  14. Numerical Experiments
  15. Equations, Fluxes, and Time Integration
  16. ...and 11 more sections

Key Result

Theorem 1.3

(Carathéodory's Theorem caratheodory-thm) Let $S \subset \mathbb{R}^m$. Any point $\bm{p} \in \mathop{\mathrm{conv}}\nolimits(S)$ (the convex hull of $S$), can be expressed as a convex combination of at most $(m+1)$ points in $S$.

Figures (13)

  • Figure 1: An example of a domain with embedded boundaries.
  • Figure 2: An example of a Cartesian cut mesh with cut elements shown in purple and yellow. Notice that for an equivalent quad-tri mesh every cut element would be decomposed into at least one curvilinear triangular element.
  • Figure 3: An example of the subtriangulation process used on each cut element mapping a subtraingle to the reference triangle $\hat{T}$.
  • Figure 4: The initial water height for the entropy residual experiments.
  • Figure 5: The entropy residual for the entropy conservative scheme without (\ref{['fig:entropy-res-EC-noSRD']}) and with (\ref{['fig:entropy-res-EC-SRD']}) state redistribution. Without state redistribution, the entropy residual is zero up to numerical round-off error.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4