Publications
Preprints
- Wasilij Barsukow:
All-speed numerical methods for the Euler equations via a triangular-implicit time integration, 2021 submitted- Abstract:This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered discretizations for the acoustic part of the Euler equations and stabilizing them with a leap-frog-type (``triangular-implicit'') time integration, implementable as an explicit method. This takes inpiration from staggered grid numerical methods. In this way, users of collocated methods are enabled to profit from the advantages of staggered methods. The paper provides a number of new collocated schemes for linear acoustic/Maxwell equations that are inspired by the Yee scheme. They are then extended to an all-speed method for the full Euler equations on Cartesian grids. By taking the opposite view and taking inspiration from collocated methods, the paper also suggests a new way of staggering the variables which increases the stability as compared to the traditional Yee scheme. The paper thus bridges -- theoretically and practically -- the low Mach number treatment in collocated and staggered methods.
- Abstract:This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered discretizations for the acoustic part of the Euler equations and stabilizing them with a leap-frog-type (``triangular-implicit'') time integration, implementable as an explicit method. This takes inpiration from staggered grid numerical methods. In this way, users of collocated methods are enabled to profit from the advantages of staggered methods. The paper provides a number of new collocated schemes for linear acoustic/Maxwell equations that are inspired by the Yee scheme. They are then extended to an all-speed method for the full Euler equations on Cartesian grids. By taking the opposite view and taking inspiration from collocated methods, the paper also suggests a new way of staggering the variables which increases the stability as compared to the traditional Yee scheme. The paper thus bridges -- theoretically and practically -- the low Mach number treatment in collocated and staggered methods.
- Wasilij Barsukow, Jonas P. Berberich:
A well-balanced Active Flux scheme for the shallow water equations with wetting and drying, 2020 submitted- Abstract: Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.
- Abstract: Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.
Refereed journal articles
- Wasilij Barsukow, Christian Klingenberg:
Exact solution and a truly multidimensional Godunov scheme for the acoustic equations, M2AN (2022) 56(1) (pdf, doi) - Wasilij Barsukow, Jonas P. Berberich, Christian Klingenberg:
On the active flux scheme for hyperbolic PDEs with source terms, SISC (2021) 43(6): A4015-A4042 (pdf, doi) - Wasilij Barsukow:
Truly multi-dimensional all-speed schemes for the Euler equations on Cartesian grids, J. Comp. Phys. 435 (2021), 110216, (pdf, doi) - Wasilij Barsukow:
The active flux scheme for nonlinear problems, J.Sci.Comp. (2021), 86 (pdf, doi) - Wasilij Barsukow, Jonathan Hohm, Christian Klingenberg, Philip L. Roe:
The active flux scheme on Cartesian grids and its low Mach number limit, J.Sci.Comp. (2019), 81(1): 594-622 (pdf, doi) - Wasilij Barsukow:
Stationarity preserving schemes for multi-dimensional linear systems, Math.Comp. (2019) 88(318): 1621-1645, (pdf, doi) - Wasilij Barsukow, Philipp V. F. Edelmann, Christian Klingenberg, Fabian Miczek, Friedrich K. Roepke:
A numerical scheme for the compressible low-Mach number regime of ideal fluid dynamics, J.Sci.Comp. (2017) 72(2): 623-646, (pdf, doi) - Marcelo M. Miller Bertolami, Maxime Viallet, Vincent Prat, Wasilij Barsukow, Achim Weiss:
On the relevance of bubbles and potential flows for stellar convection MNRAS (2016) 457 (4): 4441-4453, (pdf, doi)
Refereed conference proceedings
- Wasilij Barsukow:
Stationarity preservation properties of the active flux scheme on Cartesian grids, Proceedings of HONOM2019, Commun. Appl. Math. Comput., 2020 (doi, pdf) - Wasilij Barsukow:
Stationary states of finite volume discretizations of multi-dimensional linear hyperbolic systems, Proc. of the XVII International Conference on Hyperbolic Problems (HYP2018), A. Bressan et al. (eds), AIMS Series on Applied Mathematics Vol. 10, 2020 (pdf) - Wasilij Barsukow:
Stationarity and vorticity preservation for the linearized Euler equations in multiple spatial dimensions, Finite Volumes for Complex Applications VIII — Methods and Theoretical Aspects, C. Cancès and P. Omnes (eds.), Springer Proceedings in Mathematics & Statistics 199, 2017 (doi) - Wasilij Barsukow, Philipp V. F. Edelmann, Christian Klingenberg, Friedrich K. Roepke:
A low-Mach Roe-type solver for the Euler equations allowing for gravity source terms, Workshop on low velocity flows, Paris, 5-6 Nov. 2015, Dellacherie et al. (eds.), ESAIM: Proceedings and Surveys, Volume 56, 2017, (doi, pdf)
PhD Thesis
- Wasilij Barsukow: Low Mach number finite volume methods for the acoustic and Euler equations, 2018 (pdf)
Other publications
- Wasilij Barsukow:
Approximate evolution operators for the Active Flux method, Oberwolfach Workshop Report 2021, 19 (doi, pdf)
Posters
- Wasilij Barsukow: Stationarity preserving schemes for the linearized Euler equations in multiple spatial dimensions (pdf)
- Wasilij Barsukow, Philipp V.F. Edelmann, Christian Klingenberg, Friedrich K. Roepke: A low-Mach Roe-type solver for the Euler equations allowing for gravity source terms (pdf)
Last modified: Thu Feb 10 15:20:05 CET 2022