ENO reconstruction and ENO interpolation are stable

TL;DR

This paper proves stability properties of ENO reconstruction and interpolation, showing they preserve the sign of jumps and are bounded by underlying data, demonstrating their robustness and rigidity across various conditions.

Contribution

The paper establishes stability estimates and sign preservation properties for ENO reconstruction and interpolation of any order on non-uniform meshes.

Findings

Jump of reconstructed values has same sign as underlying cell averages

Reconstructed jumps are upper-bounded by original jumps

Properties hold for arbitrary order and non-uniform meshes

Abstract

We prove stability estimates for the ENO reconstruction and ENO interpolation procedures. In particular, we show that the jump of the reconstructed ENO pointvalues at each cell interface has the same sign as the jump of the underlying cell averages across that interface. We also prove that the jump of the reconstructed values can be upper-bounded in terms of the jump of the underlying cell averages. Similar sign properties hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on non-uniform meshes, indicate a remarkable rigidity of the piecewise-polynomial ENO procedure.