Constrained Reconstruction in MUSCL-type Finite Volume Schemes

TL;DR

This paper introduces a stabilization method for MUSCL-type finite volume schemes using inequality-constrained optimization, applicable to arbitrary grids, and demonstrates its effectiveness through numerical experiments.

Contribution

It presents a novel constrained reconstruction technique that enhances stability and accuracy of finite volume schemes on general meshes, including Cartesian grids.

Findings

Reconstruction coincides with Minmod on Cartesian meshes

Enhanced stability of second-order schemes demonstrated

Numerical experiments confirm improved accuracy

Abstract

In this paper we are concerned with the stabilization of MUSCL-type finite volume schemes in arbitrary space dimensions. We consider a number of limited reconstruction techniques which are defined in terms inequality-constrained linear or quadratic programming problems on individual grid elements. No restrictions to the conformity of the grid or the shape of its elements are made. In the special case of Cartesian meshes a novel QP reconstruction is shown to coincide with the widely used Minmod reconstruction. The accuracy and overall efficiency of the stabilized second-order finite volume schemes is supported by numerical experiments.