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

Core Concepts

The authors present a provably energy stable high-order discontinuous Galerkin (DG) method for solving the acoustic wave equation on cut meshes. They pair the DG scheme with state redistribution, a technique to address the small cell problem on cut meshes, and prove that the resulting scheme remains energy stable.

Abstract

The authors present a high-order energy stable discontinuous Galerkin (DG) method for solving the acoustic wave equation on cut meshes. The key points are: The DG formulation is derived in skew-symmetric form to ensure energy stability under arbitrary quadrature rules, which are necessary for cut elements. State redistribution is used to address the small cell problem on cut meshes by merging and redistributing the solution on small cut elements. The authors prove that state redistribution can be added to the energy stable DG scheme without damaging its energy stability. Numerical experiments are performed to verify the high-order accuracy, energy stability, and CFL condition relaxation of the scheme on 2D wave propagation problems, including a comparison to the "Pacman" benchmark.

Stats

1 c2 ∂p ∂t + ∇· u = 0 ∂u ∂t + ∇p = 0 The acoustic wave equation governing the pressure p and velocity u.

Quotes

"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." "State redistribution can be used to address the small cell problem." "We prove that state redistribution can be added to a provably L2 energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's L2 stability."

Key Insights Distilled From

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

by Christina G.... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.06630.pdf

Deeper Inquiries

How would the scheme perform on 3D problems with more complex embedded geometries?

In 3D problems with more complex embedded geometries, the scheme may face challenges due to the increased complexity of the geometry. The explicit parameterization used to represent embedded boundaries may struggle to accurately capture intricate 3D shapes with sharp features. This could lead to difficulties in determining merge neighborhoods and constructing accurate volume quadrature rules. Additionally, the computational cost of handling 3D geometries with explicit parameterizations may increase significantly, impacting the efficiency of the scheme. Overall, while the scheme may still be applicable to 3D problems, it may require enhancements and optimizations to effectively handle more complex embedded geometries.

What are the limitations of using explicit parameterizations to represent embedded boundaries compared to level-set or other implicit representations?

Using explicit parameterizations to represent embedded boundaries has certain limitations compared to level-set or other implicit representations. Some of these limitations include: Difficulty in Handling Complex Geometries: Explicit parameterizations may struggle to accurately represent complex and irregular geometries with sharp features, leading to challenges in determining intersections and constructing accurate quadrature rules. Lack of Flexibility: Explicit parameterizations are rigid and may not easily adapt to changes in the geometry, making it challenging to handle dynamic or evolving boundaries. Increased Computational Cost: The computational cost of using explicit parameterizations can be higher, especially in 3D problems with intricate geometries, as they may require more computational resources and time to process. Limited Smoothness: Explicit parameterizations may not provide the smoothness required for certain numerical computations, especially in the presence of sharp corners or edges. Negative Quadrature Weights: In some cases, explicit parameterizations can lead to negative quadrature weights, which can introduce numerical instability and inaccuracies in the calculations.

How could the quadrature construction on cut elements be improved to avoid negative weights and better condition the volume integrals?

To improve the quadrature construction on cut elements and avoid negative weights while better conditioning the volume integrals, the following strategies can be implemented: Positive Quadrature Weights: Implement methods that ensure the quadrature weights are strictly positive, such as using specialized algorithms like positive polynomial interpolation or optimization techniques to enforce positivity constraints. Improved Sampling Techniques: Utilize advanced sampling techniques to generate quadrature nodes that are well-distributed and avoid clustering, which can lead to numerical instabilities. Regularization Methods: Apply regularization techniques to the quadrature construction process to enhance the conditioning of the volume integrals and mitigate issues with negative weights. Optimization Algorithms: Employ optimization algorithms to optimize the quadrature weights while considering constraints that ensure positivity and numerical stability. Adaptive Quadrature Rules: Develop adaptive quadrature rules that adjust the node locations and weights based on the local geometry of the cut elements, ensuring accurate and stable integration over irregular shapes. By implementing these strategies, the quadrature construction on cut elements can be enhanced to produce more reliable and accurate results while avoiding negative weights and improving the conditioning of the volume integrals.