# An Integrated Quadratic Reconstruction for Finite Volume Schemes to Scalar Conservation Laws in Multiple Dimensions

## TL;DR

This paper introduces a parameter-free, integrated quadratic reconstruction method for finite volume schemes solving scalar conservation laws in multiple dimensions, achieving high accuracy and stability.

## Contribution

The paper presents a novel integrated quadratic reconstruction method that is parameter-free, applicable on flexible grids, and ensures a local maximum principle in finite volume schemes.

## Key findings

- Achieves third-order accuracy for smooth solutions in 2D and 3D.
- Satisfies a local maximum principle to prevent overshoots.
- Demonstrates effectiveness through numerical examples.

## Abstract

We proposed a piecewise quadratic reconstruction method in multiple dimensions, which is in an integrated style, for finite volume schemes to scalar conservation laws. This integrated quadratic reconstruction is parameter-free and applicable on flexible grids. We show that the finite volume schemes with the new reconstruction satisfy a local maximum principle with properly setup on time steplength. Numerical examples are presented to show that the proposed scheme attains a third-order accuracy for smooth solutions in both 2D and 3D cases. It is indicated by numerical results that the local maximum principle is helpful to prevent overshoots in numerical solutions.