On the Active Flux scheme for hyperbolic PDEs with source terms

TL;DR

This paper extends the Active Flux scheme to include source terms in hyperbolic PDEs, maintaining third order accuracy and enabling well-balanced solutions for systems like linear acoustics with gravity.

Contribution

The paper introduces a novel approach to incorporate source terms into the Active Flux scheme while preserving its third order accuracy.

Findings

Successfully extended Active Flux to source terms with nonlinear effects.

Achieved a well-balanced scheme for linear acoustics with gravity.

Maintained third order accuracy in the extended scheme.

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: the continuous reconstruction serves as initial data for the evolution of the points values. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. This paper focuses on the conceptual extension of Active Flux to include source terms, and thus for simplicity assumes the homogeneous part of the equations to be linear. To a large part, the treatment of the source terms is independent of the choice of the homogeneous part of the system. Additionally, only systems are considered which admit characteristics (instead of characteristic cones). This is the case for scalar equations in any number of spatial dimensions and systems in one spatial dimension. Here, we succeed to extend the Active Flux method to include (possibly nonlinear) source terms while maintaining third order accuracy of the method. This requires a novel (approximate) operator for the evolution of point values and a modified update procedure of the cell average. For linear acoustics with gravity, it is shown how to achieve a well-balanced / stationarity preserving numerical method.