A semi-discrete Active Flux method for the Euler equations on Cartesian grids

TL;DR

This paper introduces a semi-discrete Active Flux method for solving the Euler equations on Cartesian grids, achieving high-order accuracy without the need for complex evolution operators, and demonstrates its effectiveness on various fluid dynamics problems.

Contribution

It presents a novel semi-discrete Active Flux method for multi-dimensional nonlinear hyperbolic systems that avoids the use of evolution operators, enhancing computational efficiency and applicability.

Findings

Successfully applied to Riemann problems and subsonic flows.

Achieves third-order accuracy in multiple dimensions.

Demonstrates improved performance over fully discrete methods.

Abstract

Active Flux is an extension of the Finite Volume method and additionally incorporates point values located at cell boundaries. This gives rise to a globally continuous approximation of the solution. Originally, the Active Flux method emerged as a fully discrete method, and required an exact or approximate evolution operator for the point value update. For nonlinear problems such an operator is often difficult to obtain, in particular for multiple spatial dimensions. We demonstrate that a new semi-discrete Active Flux method (first described in Abgrall&Barsukow, 2023 for one space dimension) can be used to solve nonlinear hyperbolic systems in multiple dimensions without requiring evolution operators. We focus here on the compressible Euler equations of inviscid hydrodynamics and third-order accuracy. We introduce a multi-dimensional limiting strategy and demonstrate the performance of the new method on both Riemann problems and subsonic flows.