Analysis of the Multi-dimensional Semi-discrete Active Flux Method Using the Fourier Transform

Abstract

The degrees of freedom of Active Flux are cell averages and point values along the cell boundaries. The latter are shared between neighboring cells, which gives rise to a globally continuous reconstruction. The semi-discrete Active Flux method uses its degrees of freedom to obtain Finite Difference approximations 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.

Keywords

Active flux / Stationarity-preserving / Linear acoustics / Fourier transform / 65M20 / 65M70 / 65M08 / 35E15

References

[1] AbgrallR. A combination of residual distribution and the active flux formulations or a new class of schemes that can combine several writings of the same hyperbolic problem: application to the 1D Euler equations. Commun. Appl. Math. Comput., 2023, 5: 370-402.

[2] AbgrallR, BarsukowW. Extensions of active flux to arbitrary order of accuracy. ESAIM Math. Model. Numer. Anal., 2023, 57(2): 991-1027.

[3] AbgrallR, BarsukowW. A hybrid finite element-finite volume method for conservation laws. Appl. Math. Comput., 2023, 447127846

[4] AbgrallR, BarsukowW, KlingenbergC. A semi-discrete active flux method for the Euler equations on Cartesian grids. J. Sci. Comput., 2025, 102236.

[5] Barsukow, W.: Low Mach number finite volume methods for the acoustic and Euler equations. Doctoral thesis, University of Wuerzburg (2018)

[6] BarsukowW. Stationarity preserving schemes for multi-dimensional linear systems. Math. Comput., 2019, 88(318): 1621-1645.

[7] BarsukowW. The active flux scheme for nonlinear problems. J. Sci. Comput., 2021, 86(1): 1-34.

[8] BarsukowW. Stationarity preservation properties of the active flux scheme on cartesian grids. Commun. Appl. Math. Comput., 2023, 5(2): 638-652.

[9] BarsukowW, HohmJ, KlingenbergC, RoePL. The active flux scheme on Cartesian grids and its low Mach number limit. J. Sci. Comput., 2019, 81(1): 594-622.

[10] BarsukowW, KlingenbergC. Exact solution and the multidimensional Godunov scheme for the acoustic equations. ESAIM Math. Model. Numer. Anal., 2022, 56(1): 317-347.

[11] BarsukowW, LoubereR, MaireP-H. A node-conservative vorticity-preserving finite volume method for linear acoustics on unstructured grids. Math. Comput., 2023, 94(355): 2299-2343.

[12] ChudzikE, HelzelC, KerkmannD. The Cartesian grid active flux method: linear stability and bound preserving limiting. Appl. Math. Comput., 2021, 393125501

[13] Eymann, T.A., Roe, P.L.: Multidimensional active flux schemes. In: 21st AIAA Computational Fluid Dynamics Conference, AIAA 2013-2940 (2013)

[14] HelzelC, KerkmannD, ScandurraL. A new ADER method inspired by the active flux method. J. Sci. Comput., 2019, 80(3): 1463-1497.

[15] MortonKW, RoePL. Vorticity-preserving Lax-Wendroff-type schemes for the system wave equation. SIAM J. Sci. Comput., 2001, 23(1): 170-192.

[16] RoeP. Designing CFD methods for bandwidth—a physical approach. Comput. Fluids, 2021, 214104774.

[17] van LeerB. Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. J. Comput. Phys., 1977, 23(3): 276-299.

[18] ZengX. Linear hybrid-variable methods for advection equations. Adv. Comput. Math., 2019, 45(2): 929-980.

Funding

DFG(525941602)

RIGHTS & PERMISSIONS

Shanghai University

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