# On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws

## TL;DR

This paper develops a third order CWENO boundary treatment for hyperbolic conservation law networks, enhancing stability and accuracy at domain boundaries, with numerical validation of the theoretical bounds.

## Contribution

It refines and generalizes third order boundary treatment techniques for hyperbolic conservation laws, extending previous WENO-based methods.

## Key findings

- Numerical evidence supports the analytically derived parameter bounds.
- A complete third order scheme demonstrates the effectiveness of the boundary treatment.
- The method achieves high order accuracy while maintaining stability near discontinuities.

## Abstract

High order numerical methods for networks of hyperbolic conservation laws have recently gained increasing popularity. Here, the crucial part is to treat the boundaries of the single (one-dimensional) computational domains in such a way that the desired convergence rate is achieved in the smooth case but also stability criterions are fulfilled, in particular in the presence of discontinuities. Most of the recently proposed methods rely on a WENO extrapolation technique introduced by Tan and Shu in [\emph{J.\ Comput.\ Phys.} 229, pp.\ 8144--8166 (2010)]. Within this work, we refine and in a sense generalize these results for the case of a third order scheme. Numerical evidence for the analytically found parameter bounds is given as well as results for a complete third order scheme based on the proposed boundary treatment.