A note on the general multi-moment constrained flux reconstruction formulation for high order schemes

TL;DR

This paper introduces a flexible multi-moment constrained flux reconstruction framework for high order schemes, extending existing methods by incorporating diverse constraints on flux functions and derivatives, enhancing scheme design options.

Contribution

It proposes a novel general formulation (MMC-FR) that broadens high order scheme construction by including various constraint types beyond flux continuity.

Findings

Framework accommodates a wider class of high order schemes.

Fourier analysis validates the stability and accuracy of the proposed schemes.

Numerical tests demonstrate improved performance over traditional methods.

Abstract

This paper presents a general formulation to construct high order numerical schemes by using multi-moment constraint conditions on the flux function reconstruction. The new formulation, so called multi-moment constrained flux reconstruction (MMC-FR), distinguishes itself essentially from the flux reconstruction formulation (FR) of Huynh (2007) by imposing not only the continuity constraint conditions on the flux function at the cell boundary, but also other types constraints which may include those on the spatial derivatives or the point values. This formulation can be also interprated as a blend of Lagrange interpolation the Hermite interpolation, which provides a numerical framework to accomodate a wider spectrum of high order schemes. Some representative schemes will be presented and evaluated through Fourier analysis and numerical tests.