The Active Flux scheme for nonlinear problems

TL;DR

This paper introduces an advanced Active Flux finite volume scheme for nonlinear hyperbolic problems, featuring approximate evolution operators, an entropy fix, and a new limiting strategy, demonstrating high accuracy on diverse test cases.

Contribution

It develops approximate evolution operators for nonlinear problems, incorporates an entropy fix and a novel limiting strategy, enhancing the Active Flux scheme's robustness and accuracy.

Findings

Achieves third-order accuracy without Riemann solvers.

Effectively handles smooth and discontinuous solutions.

Improves stability with the new entropy fix and limiting strategy.

Abstract

The Active Flux scheme is a finite volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver. Instead, given a reconstruction, the initial value problem at the location of the point value is solved. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. Whereas for linear problems an exact evolution operator is available, for nonlinear problems one needs to resort to approximate evolution operators. This paper presents such approximate operators for nonlinear hyperbolic systems in one dimension and nonlinear scalar equations in multiple spatial dimensions. They are obtained by estimating the wave speeds to sufficient order of accuracy. Additionally, an entropy fix is introduced and a new limiting strategy is proposed. The abilities of the scheme are assessed on a variety of smooth and discontinuous setups.