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

Abstract

We proposed a piecewise quadratic reconstruction method in multiple di-mensions, which is in an integrated style, for finite volume schemes to scalar conser-vation 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. Numer-ical 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.

Keywords

Quadratic reconstruction / finite volume method / local maximum principle / scalar conservation law / unstructured mesh