# Compact Approximate Taylor methods for systems of conservation laws

## TL;DR

This paper introduces the Compact Approximate Taylor (CAT) methods, high-order schemes for conservation laws that are stable, accurate, and capable of handling discontinuities with shock-capturing techniques, showing promising results in various test cases.

## Contribution

The paper develops a new family of high-order, linearly stable CAT methods based on centered stencils, extending Lax-Wendroff methods and incorporating shock-capturing techniques.

## Key findings

- CAT methods achieve order 2p accuracy with CFL<1 stability.
- WENO-CAT methods effectively prevent oscillations near discontinuities.
- CAT methods perform well on test cases like Burgers and Euler equations.

## Abstract

A new family of high order methods for systems of conservation laws are introduced: the Compact Approximate Taylor (CAT) methods. These methods are based on centered (2p + 1)-point stencils where p is an arbitrary integer. We prove that the order of accuracy is 2p and that CAT methods are an extension of high-order Lax-Wendroff methods for linear problems. Due to this, they are linearly L2-stable under a CFL<1 condition. In order to prevent the spurious oscillations that appear close to discontinuities two shock-capturing techniques have been considered: a fux-limiter technique (FL-CAT methods) and WENO reconstruction for the frst time derivative (WENO-CAT methods). We follow the WENO-Lax Wendroff Approximate Taylor method of Zorio, Baeza and Mullet (2017) in the second approach. A number of test cases are considered to compare these methods with other WENO-based schemes: the linear transport equation, Burgers equation, and the 1D compressible Euler system are considered. Although CAT methods present an extra computational cost due to the local character, this extra cost is compensated by the fact that they still give good solutions with CFL values close to 1.