Relations between WENO3 and Third-order Limiting in Finite Volume Methods

TL;DR

This paper presents a unified framework linking WENO3 and third-order finite volume limiters by generalizing limiter functions to a two-dimensional slope dependence, aiming to improve limiting strategies.

Contribution

It introduces a generalized limiter function perspective that unifies WENO3 and FV schemes, revealing limitations of classical FV limiters and opening avenues for enhancements.

Findings

Unified view of WENO3 and FV limiters through 2D slope dependence

Classical FV limiters are overly restrictive in their approach

Potential for improved limiting strategies in both methods

Abstract

Weighted essentially non-oscillatory (WENO) and finite volume (FV) methods employ different philosophies in their way to perform limiting. We show that a generalized view on limiter functions, which considers a two-dimensional, rather than a one-dimensional dependence on the slopes in neighboring cells, allows to write WENO3 and $3^\text{rd}$-order FV schemes in the same fashion. Within this framework, it becomes apparent that the classical approach of FV limiters to only consider ratios of the slopes in neighboring cells, is overly restrictive. The hope of this new perspective is to establish new connections between WENO3 and FV limiter functions, which may give rise to improvements for the limiting behavior in both approaches.