The Cartesian Grid Active Flux Method with Adaptive Mesh Refinement

TL;DR

This paper introduces the first implementation of the third order accurate Active Flux method on adaptively refined Cartesian grids, enhancing efficiency and stability for solving hyperbolic conservation laws.

Contribution

The paper develops and implements an adaptive mesh refinement version of the Active Flux method within ForestClaw, enabling efficient, stable, and high-order accurate solutions on complex grids.

Findings

Achieves third order accuracy on adaptive grids

Demonstrates efficient data exchange via ghost cells

Supports subcycling in time for improved performance

Abstract

We present the first implementation of the Active Flux method on adaptively refined Cartesian grids. The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method has a compact stencil in space and time and good stability properties. The method is implemented as a new solver in ForestClaw, a software for parallel adaptive mesh refinement of patch-based solvers. On each Cartesian grid patch the single grid Active Flux method can be applied. The exchange of data between grid patches is organised via ghost cells. The local stencil in space and time and the availability of the point values that are used for the reconstruction, leads to an efficient implementation. The resulting method is third order accurate, conservative and allows the use of subcycling in time.