A hybrid finite element - finite volume method for conservation laws

TL;DR

This paper introduces a high-order accurate numerical method combining finite element and finite volume techniques for conservation laws, utilizing point values and moments for improved solution approximation.

Contribution

It presents a novel hybrid method that unifies finite element and finite volume approaches with high-order accuracy and flexible point value updates.

Findings

Method reduces to Active Flux at third order.

Stability and accuracy are thoroughly analyzed.

Two update strategies for point values are proposed.

Abstract

We propose an arbitrarily high-order accurate numerical method for conservation laws that is based on a continuous approximation of the solution. The degrees of freedom are point values at cell interfaces and moments of the solution inside the cell. To lowest ($3^\text{rd}$) order this method reduces to the Active Flux method. The update of the moments is achieved immediately by integrating the conservation law over the cell, integrating by parts and employing the continuity across cell interfaces. We propose two ways how the point values can be updated in time: either by first deriving a semi-discrete method that uses a finite-difference-type formula to approximate the spatial derivative, and integrating this method e.g. with a Runge-Kutta scheme, or by using a characteristics-based update, which is inspired by the original (fully discrete) Active Flux method. We analyze stability and accuracy of the resulting methods.