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Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform

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arxiv 2412.03477 v2 pith:HK4GWEWH submitted 2024-12-04 math.NA cs.NA

Wasilij Barsukow , Janina Kern , Christian Klingenberg , Lisa Lechner This is my paper · ORCID
classification math.NAcs.NA
keywords fluxactivemethodpointanalysisaveragescelldegrees
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The degrees of freedom of Active Flux are cell averages and point values along the cell boundaries. These latter are shared between neighbouring cells, which gives rise to a globally continuous reconstruction. The semi-discrete Active Flux method uses its degrees of freedom to obtain Finite Difference approxi\-mations to the spatial derivatives which are used in the point value update. The averages are updated using a quadrature of the flux and making use of the point values as quadrature points. The integration in time employs standard Runge-Kutta methods. We show that this generalization of the Active Flux method in two and three spatial dimensions is stationarity preserving for linear acoustics on Cartesian grids, and present an analysis of numerical diffusion and stability.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An Active Flux method for the Euler equations based on the exact acoustic evolution operator

    math.NA 2025-06 conditional novelty 7.0 of 10

    A new Active Flux scheme for the 2D Euler equations combines an exactly solved locally linearized acoustic operator with a third-order advective evolution operator inside an additive splitting, using primitive variabl...

  2. Semi-discrete Active Flux as a Petrov-Galerkin method: the case of one-dimensional and Cartesian grids

    math.NA 2025-08 unverdicted novelty 6.0 of 10

    Semi-discrete Active Flux methods are shown to be Petrov-Galerkin schemes with discontinuous biorthogonal test functions, with explicit constructions for arbitrary order in one dimension and third order on Cartesian grids.

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