Interpolation and approximation of piecewise smooth functions with corner discontinuities on sigma quasi-uniform grids

TL;DR

This paper analyzes nonlinear interpolation methods, specifically ENO and SR techniques, on quasi-uniform grids to accurately approximate functions with corner discontinuities, with proven approximation orders.

Contribution

It establishes approximation orders for nonlinear interpolation methods on quasi-uniform grids, including singularity detection and optimal approximation for certain nonuniform grids.

Findings

Optimal approximation order is achieved for grids with maximum spacing below a critical value.

The methods effectively detect isolated singularities in functions.

Approximation capabilities are proven for functions with corner singularities.

Abstract

This paper provides approximation orders for a class of nonlinear interpolation procedures for univariate data sampled over $\sigma$ quasi-uniform grids. The considered interpolation is built using both essentially nonoscillatory (ENO) and subcell resolution (SR) reconstruction techniques. The main target of these nonlinear techniques is to reduce the approximation error for functions with isolated corner singularities and in turn this fact makes them useful for applications to other fields, such as shock capturing computations or image processing. We start proving the approximation capabilities of an algorithm to detect the presence of isolated singularities, and then we address the approximation order attained by the mentioned interpolation procedure. For certain nonuniform grids with a maximum spacing between nodes $h$ below a critical value $h_c$, the optimal approximation order is recovered, as it happens for uniformly smooth functions \cite{ACDD}.