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

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.