A Bound Preserving Cut Discontinuous Galerkin Method for One-Dimensional Hyperbolic Conservation Laws

Core Concepts

The authors present a family of high-order cut finite element methods with bound preserving properties for one-dimensional hyperbolic conservation laws. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. The proposed schemes ensure conservation, optimal order of accuracy, and bound preservation through a reconstruction on macro-elements and the application of suitable limiters.

Abstract

The paper presents a family of high-order cut discontinuous Galerkin (Cut-DG) methods for solving one-dimensional hyperbolic conservation laws. The key aspects are:

The methods use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. This creates small cut elements that can cause numerical difficulties.

To handle the small cut elements, the authors use ghost penalty stabilization and a reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization.

For scalar conservation laws, the authors show that the piecewise constant scheme satisfies a maximum principle under a suitable time step restriction. For higher-order schemes, they prove that the mean values of the reconstructed solutions on macro-elements satisfy the corresponding bounds.

To ensure bound preservation for the high-order schemes, the authors apply suitable bound preserving limiters to the reconstructed solutions on macro-elements.

For the Euler equations, the authors show that the proposed schemes are positivity preserving in the sense that positivity of pressure and density are retained.

The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh.

Numerical experiments illustrate the accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.

Stats

The paper does not contain any specific numerical data or metrics to extract. The focus is on the theoretical analysis and development of the numerical scheme.

Quotes

None.

Key Insights Distilled From

A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws

by Pei Fu,Gunil... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13936.pdf

Deeper Inquiries

How can the proposed reconstruction and limiting techniques be extended to higher spatial dimensions

The proposed reconstruction and limiting techniques can be extended to higher spatial dimensions by following a similar approach as in the one-dimensional case. In higher dimensions, the domain will be divided into subdomains with arbitrary interfaces cutting through the computational mesh. The macro-element partitioning can still be applied to create larger elements that encompass both cut and uncut neighboring elements. The reconstruction process can then be performed on these macro-elements to ensure conservation and accuracy of the solution. Additionally, the bound preserving limiter can be applied to the reconstructed solution on each macro-element to maintain the maximum principle property. By extending these techniques to higher dimensions, the bound preserving Cut-DG method can be effectively implemented for hyperbolic conservation laws in multi-dimensional spaces.

What are the potential challenges in applying the bound preserving Cut-DG method to more complex systems of hyperbolic conservation laws beyond the Euler equations

Applying the bound preserving Cut-DG method to more complex systems of hyperbolic conservation laws beyond the Euler equations may pose several challenges. One challenge is the increased complexity of the systems, which may involve multiple equations with coupled variables and non-linear interactions. Ensuring conservation, accuracy, and stability in such systems can be more challenging than in simpler systems. Additionally, the presence of shocks, discontinuities, and complex geometries can complicate the reconstruction and limiting processes. Implementing the method for systems with varying speeds of propagation, different eigenvalues, and non-convex flux functions may require additional modifications and careful consideration to maintain the desired properties of the numerical scheme.

Can the ideas behind the macro-element based reconstruction be used to develop efficient solvers for other types of partial differential equations with complex geometries

The ideas behind the macro-element based reconstruction can be utilized to develop efficient solvers for other types of partial differential equations with complex geometries. By partitioning the domain into macro-elements that encompass both cut and uncut regions, the reconstruction process can be applied to ensure conservation and accuracy of the solution. This approach can be extended to problems with irregular boundaries, material interfaces, and intricate geometries. By incorporating bound preserving limiters and stabilization techniques, the macro-element based reconstruction can help in developing robust and accurate solvers for a wide range of partial differential equations, including those with complex geometries and challenging boundary conditions. The method can be adapted to handle different types of equations, such as diffusion equations, reaction-diffusion systems, and fluid dynamics equations, providing a versatile framework for numerical simulations in various scientific and engineering applications.