Applications of Limiters, Neural Networks and Polynomial Annihilation in Higher-Order FD/FV Schemes

The construction of high-order structure-preserving numerical schemes to solve hyperbolic conservation laws has attracted a lot of attention in the last decades and various different ansatzes exist. In this paper, we compare several completely different approaches, i.e. deep neural networks, limiters and the application of polynomial annihilation to construct high-order accurate shock capturing finite difference/volume (FD/FV) schemes. We further analyze their analytical and numerical properties. We demonstrate that all techniques can be used and yield highly efficient FD/FV methods but also come with some additional drawbacks which we point out. Our investigation of the different strategies should lead to a better understanding of those techniques and can be transferred to other numerical methods as well which use similar ideas.

Automated summary

The construction of high-order structure-preserving numerical schemes for hyperbolic conservation laws has been extensively studied. This paper compares different approaches, including deep neural networks, limiters, and polynomial annihilation, for constructing high-order accurate shock capturing FD/FV schemes. The analytical and numerical properties of these schemes are further analyzed. The investigation of these strategies aims to enhance the understanding of these techniques and can be applied to other numerical methods with similar ideas.