Comparison to control oscillations in high-order Finite Volume schemes via physical constraint limiters, neural networks and polynomial annihilation

TL;DR

This paper compares three different approaches—physical constraint limiters, neural networks, and polynomial annihilation—for constructing high-order oscillation-free finite volume schemes for hyperbolic conservation laws, analyzing their properties and effectiveness.

Contribution

It provides a comprehensive comparison of three novel high-order FV blending strategies, highlighting their advantages and limitations for structure-preserving schemes.

Findings

All techniques produce efficient oscillation-free FV methods.

Each approach has specific drawbacks that are discussed.

Insights can be transferred to other numerical methods using similar ideas.

Abstract

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 three completely different approaches, i.e. physical constraint limiting, deep neural networks and the application of polynomial annihilation to construct high-order oscillation free Finite Volume (FV) blending schemes. We further analyze their analytical and numerical properties. We demonstrate that all techniques can be used and yield highly efficient FV methods but also come with some additional drawbacks which we point out. Our investigation of the different blending strategies should lead to a better understanding of those techniques and can be transferred to other numerical methods as well which use similar ideas.