# Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids

## TL;DR

This paper introduces an improved integrated linear reconstruction method for finite volume schemes on arbitrary unstructured grids, removing previous geometric restrictions and ensuring positivity-preserving properties.

## Contribution

The paper presents a new convex quadratic programming-based reconstruction approach that satisfies the local maximum principle on arbitrary unstructured meshes without geometric constraints.

## Key findings

- Ensures local maximum principle on arbitrary unstructured grids.
- Reconstruction is parameter-free and efficient.
- Finite volume scheme remains positivity-preserving for Euler equations.

## Abstract

In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172--1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to enforce a local maximum principle is too restrictive to be satisfied by, for example, locally refined meshes or distorted meshes generated by arbitrary Lagrangian-Eulerian methods in practical applications. In this paper, we propose an improved integrated linear reconstruction approach to get rid of the geometric hypothesis. The resulting optimization problem is a convex quadratic programming problem, and hence can be solved efficiently by classical active-set methods. The features of the improved integrated linear reconstruction include that i). the local maximum principle is fulfilled on arbitrary unstructured grids, ii). the reconstruction is parameter-free, and iii). the finite volume scheme is positivity-preserving when the reconstruction is generalized to the Euler equations. A variety of numerical experiments are presented to demonstrate the performance of this method.