Publications
Lists of publications on preprint servers: arXiv, hal
Preprints
- W. Barsukow, P. Chandrashekar, C. Klingenberg, L. Lechner:
A generalized Active Flux method of arbitrarily high order in two dimensions, 2025 submitted (pdf)- Abstract:The Active Flux method can be seen as an extended finite volume method. The degrees of freedom of this method are cell averages, as in finite volume methods, and in addition shared point values at the cell interfaces, giving rise to a globally continuous reconstruction. Its classical version was introduced as a one-stage fully discrete, third-order method. Recently, a semi-discrete version of the Active Flux method was presented with various extensions to arbitrarily high order in one space dimension.
In this paper we extend the semi-discrete Active Flux method on two-dimensional Cartesian grids to arbitrarily high order, by including moments as additional degrees of freedom (hybrid finite element--finite volume method). The stability of this method is studied for linear advection. For a fully discrete version, using an explicit Runge-Kutta method, a CFL restriction is derived. We end by presenting numerical examples for hyperbolic conservation laws.
- Abstract:The Active Flux method can be seen as an extended finite volume method. The degrees of freedom of this method are cell averages, as in finite volume methods, and in addition shared point values at the cell interfaces, giving rise to a globally continuous reconstruction. Its classical version was introduced as a one-stage fully discrete, third-order method. Recently, a semi-discrete version of the Active Flux method was presented with various extensions to arbitrarily high order in one space dimension.
- W. Barsukow, M. Ricchiuto, D. Torlo:
Structure preserving nodal continuous Finite Elements via Global Flux quadrature, 2024 submitted (pdf)- Abstract:Numerical methods for hyperbolic PDEs require stabilization. For linear acoustics, divergence-free vector fields should remain stationary, but classical Finite Difference methods add incompatible diffusion that dramatically restricts the set of discrete stationary states of the numerical method. Compatible diffusion should vanish on stationary states, e.g. should be a gradient of the divergence. Some Finite Element methods allow to naturally embed this grad-div structure, e.g. the SUPG method or OSS. We prove here that the particular discretization associated to them still fails to be constraint preserving. We then introduce a new framework on Cartesian grids based on surface (volume in 3D) integrated operators inspired by Global Flux quadrature and related to mimetic approaches. We are able to construct constraint-compatible stabilization operators (e.g. of SUPG-type) and show that the resulting methods are vorticity-preserving. We show that the Global Flux approach is even super-convergent on stationary states, we characterize the kernels of the discrete operators and we provide projections onto them.
- Abstract:Numerical methods for hyperbolic PDEs require stabilization. For linear acoustics, divergence-free vector fields should remain stationary, but classical Finite Difference methods add incompatible diffusion that dramatically restricts the set of discrete stationary states of the numerical method. Compatible diffusion should vanish on stationary states, e.g. should be a gradient of the divergence. Some Finite Element methods allow to naturally embed this grad-div structure, e.g. the SUPG method or OSS. We prove here that the particular discretization associated to them still fails to be constraint preserving. We then introduce a new framework on Cartesian grids based on surface (volume in 3D) integrated operators inspired by Global Flux quadrature and related to mimetic approaches. We are able to construct constraint-compatible stabilization operators (e.g. of SUPG-type) and show that the resulting methods are vorticity-preserving. We show that the Global Flux approach is even super-convergent on stationary states, we characterize the kernels of the discrete operators and we provide projections onto them.
- W. Barsukow, Y. Liu:
An Arbitrarily High-Order Fully Well-balanced Hybrid Finite Element-Finite Volume Method for a One-dimensional Blood Flow Model, 2024 submitted (pdf)- Abstract: In this paper, we propose an arbitrarily high-order accurate fully well-balanced numerical method for the one-dimensional blood flow model. The developed method is based on a continuous representation of the solution and a natural combination of the conservative and primitive formulations of the studied PDEs. The degrees of freedom are defined as point values at cell interfaces and moments of the conservative variables inside the cell, drawing inspiration from the discontinuous Galerkin method. The well-balanced property, in the sense of an exact preservation of both the zero and non-zero velocity equilibria, is achieved by a well-balanced approximation of the source term in the conservative formulation and a well-balanced residual computation in the primitive formulation. To lowest (3rd) order this method reduces to the method developed in [Abgrall and Liu, A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations, arXiv preprint, arXiv:2304.07809]. Several numerical tests are shown to prove its well-balanced and high-order accuracy properties.
- Abstract: In this paper, we propose an arbitrarily high-order accurate fully well-balanced numerical method for the one-dimensional blood flow model. The developed method is based on a continuous representation of the solution and a natural combination of the conservative and primitive formulations of the studied PDEs. The degrees of freedom are defined as point values at cell interfaces and moments of the conservative variables inside the cell, drawing inspiration from the discontinuous Galerkin method. The well-balanced property, in the sense of an exact preservation of both the zero and non-zero velocity equilibria, is achieved by a well-balanced approximation of the source term in the conservative formulation and a well-balanced residual computation in the primitive formulation. To lowest (3rd) order this method reduces to the method developed in [Abgrall and Liu, A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations, arXiv preprint, arXiv:2304.07809]. Several numerical tests are shown to prove its well-balanced and high-order accuracy properties.
- W. Barsukow, J. Kern, C. Klingenberg, L. Lechner:
Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform, 2024 submitted (pdf)- Abstract: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 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.
- Abstract: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 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.
Refereed journal articles
- J. Duan, W. Barsukow, C. Klingenberg:
Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation, 2024 accepted in SISC (pdf, combines 2405.02447 (1D) and 2407.13380 (multi-D)) - Remi Abgrall, Wasilij Barsukow, Christian Klingenberg:
A semi-discrete Active Flux method for the Euler equations on Cartesian grids, 2024 accepted in J. Sci. Comp. (pdf, hal) - Wasilij Barsukow, Raphaël Loubère, Pierre-Henri Maire:
A node-conservative vorticity-preserving Finite Volume method for linear acoustics on unstructured grids, 2024 accepted in Math. Comp. (pdf) - G. Leidi, R. Andrassy, W. Barsukow, J. Higl, P. V. F. Edelmann, F. K. Röpke:
Performance of high-order Godunov-type methods in simulations of astrophysical low Mach number flows, A&A 686 (2024) A34 (pdf, doi) - Wasilij Barsukow, Raul Borsche:
Implicit Active Flux methods for linear advection, J. Sci. Comp. (2024) 98(3) (pdf, doi) - Wasilij Barsukow:
All-speed numerical methods for the Euler equations via a sequential explicit time integration, J.Sci.Comp. (2023), 95 (pdf, doi) - Remi Abgrall, Wasilij Barsukow:
Extensions of Active Flux to arbitrary order of accuracy, M2AN (2023) 57(2): 991-1027 (pdf, hal, doi) - Wasilij Barsukow, Jonas P. Berberich:
A well-balanced Active Flux scheme for the shallow water equations with wetting and drying, CAMC (2023): 1-46 (pdf, hal, doi) - Wasilij Barsukow, Christian Klingenberg:
Exact solution and a truly multidimensional Godunov scheme for the acoustic equations, M2AN (2022) 56(1): 317-347 (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
- Alessia Del Grosso, Wasilij Barsukow, Raphaël Loubère, Pierre-Henri Maire:
An asymptotic-preserving multidimensionality-aware finite volume numerical scheme for Euler equations, 2024 accepted as proceedings of ICCFD12 (hal) - Wasilij Barsukow:
Truly multi-dimensional all-speed methods for the Euler equations, Proc. of FVCA10, Springer Proceedings 2023, pp. 23-31. (pdf, doi) - Remi Abgrall, Wasilij Barsukow:
A hybrid finite element-finite volume method for conservation laws, Proc. of the NumHyp21 conference, AMC 447 (2023): 127846 (pdf, hal, doi) - Wasilij Barsukow:
Stationarity preservation properties of the active flux scheme on Cartesian grids, Proc. 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:
Preserving stationary states on unstructured grids, Oberwolfach Workshop Report 2024 - Wasilij Barsukow:
Time integration of the semi-discrete Active Flux method, Oberwolfach Workshop Report 2022, 19 - Wasilij Barsukow:
Approximate evolution operators for the Active Flux method, Oberwolfach Workshop Report 2021, 19 (doi, pdf)
Posters
- Wasilij Barsukow: Truly multi-dimensional all-speed methods for the Euler equations (pdf)
- 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: Mon Feb 10 08:17:35 CET 2025