# A generalized framework to construct third order WENO weights using weight limiter functions

## TL;DR

This paper introduces a novel, generalized framework for constructing third order WENO weights using weight limiter functions, ensuring non-oscillatory solutions and third order accuracy, including at smooth extrema.

## Contribution

A new framework for WENO weights based on weight limiter functions that guarantees third order accuracy and non-oscillatory behavior, distinct from existing methods.

## Key findings

- New weights guarantee third order accuracy in smooth regions.
- Proposed weights maintain non-oscillatory shock capturing.
- Numerical tests confirm effectiveness across various problems.

## Abstract

The main aim of this work is not to improve any existing non-linear weight but to give a generalized framework for the construction of non-linear weights to get non-oscillatory third order WENO schemes. It is done by imposing necessary conditions on weights to get non-oscillatory WENO reconstruction which give further insight on the structure of weights to ensure non-occurrence of oscillations and characterize the solution region for third order accuracy. This framework for WENO weights is new and completely different from the prevailing existing approach. New non-linear weights are designed using a function of smoothness parameter termed as weight limiter functions. Many such weight limiter functions are given and analyzed. These new weights are simple and by construction guarantee for exact third order accuracy in smooth solution region including smooth extrema away from critical point. Numerical results for various test problems are given and compared. Results show that proposed weights give third order accuracy without loosing the non-oscillatory shock capturing ability of the resulting scheme.