New Paper ‘Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators’ published in BIT Numerical Mathematics

The paper Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators of Wasilij Barsukow, Christian Klingenberg, Lisa Lechner, Jan Nordström, Sigrun Ortleb, and me has been published in BIT Numerical Mathematics.

The Active Flux method is a numerical method for conservation laws using a combination of cell averages and point values as independent degrees of freedom, based on ideas from finite volumes and finite differences. This unusual mix has been shown to work well in many situations. We expand the theoretical justifications of the Active Flux method by analyzing it from the point of view of summation-by-parts (SBP) operators, which are routinely used to analyze finite difference, finite volume, and finite element schemes. We investigate in what type of setting the Active Flux method can be formulated using classical or degenerate SBP operators, yielding a first and novel approach for showing the energy stability of the Active Flux method. We present the analysis for the one-dimensional scalar linear advection equation with periodic boundary conditions on a uniform grid.

The reproducibility repository is available on GitHub and Zenodo.

As usual, you can find the preprint on arXiv.