Publications
Lists of publications on preprint servers: arXiv, hal
Preprints
- W. Barsukow, C. Klingenberg, S. Krotsch:
On the equivalence of semi-discrete Active Flux and Discontinuous Galerkin methods and a comparison of their performance, 2026 submitted (pdf)- Abstract:The Active Flux (AF) method employs a globally continuous approximation, like continuous Finite Element methods. This is achieved through the placement of point values at cell interfaces which are shared between adjacent cells. With, on average, K+1 degrees of freedom per cell, Active Flux achieves a polynomial approximation of degree K+1, while the Discontinuous Galerkin (DG) method uses only polynomials of degree K, i.e. one degree less with the same number of degrees of freedom. Despite all the differences, in this paper we show, however, that for linear problems in one and several dimensions as well as — in some sense — for nonlinear ones, semi-discrete AF and DG are the same method. We identify a mapping between their respective degrees of freedom, upon which the updates of these degrees of freedom turn out to agree. On the one hand, AF therefore seems more economical then DG for a given value of the error, and we confirm this in numerical experiments. On the other hand, this is a way to understand superconvergence of DG in a natural way, and we show how Radau polynomials and their zeros appear in the mapping between DG and AF: In the Radau points, AF "shines through" as the background high-order scheme behind DG.
- Abstract:The Active Flux (AF) method employs a globally continuous approximation, like continuous Finite Element methods. This is achieved through the placement of point values at cell interfaces which are shared between adjacent cells. With, on average, K+1 degrees of freedom per cell, Active Flux achieves a polynomial approximation of degree K+1, while the Discontinuous Galerkin (DG) method uses only polynomials of degree K, i.e. one degree less with the same number of degrees of freedom. Despite all the differences, in this paper we show, however, that for linear problems in one and several dimensions as well as — in some sense — for nonlinear ones, semi-discrete AF and DG are the same method. We identify a mapping between their respective degrees of freedom, upon which the updates of these degrees of freedom turn out to agree. On the one hand, AF therefore seems more economical then DG for a given value of the error, and we confirm this in numerical experiments. On the other hand, this is a way to understand superconvergence of DG in a natural way, and we show how Radau polynomials and their zeros appear in the mapping between DG and AF: In the Radau points, AF "shines through" as the background high-order scheme behind DG.
- W. Barsukow:
Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids, 2025 submitted (pdf)- Abstract:Due to added numerical stabilization (diffusion), the stationary states of numerical methods for hyperbolic problems need not be consistent discretizations of those of the PDEs. A closely related phenomenon is the lack of consistency of common finite volume methods for the Euler equations in the limit of low Mach number. In this work, the stationary states of the Discontinuous Galerkin (DG) method for linear acoustics on Cartesian grids are explored theoretically and experimentally, thus extending previous studies in the context of first-order finite difference methods. It is found that for a polynomial degree above some threshold, DG is stationarity preserving, but depending on the choice of numerical flux can suffer from a reduction of the order of accuracy at stationary state. This allows to explain the behaviour of the method for the Euler equations at low Mach number.
- Abstract:Due to added numerical stabilization (diffusion), the stationary states of numerical methods for hyperbolic problems need not be consistent discretizations of those of the PDEs. A closely related phenomenon is the lack of consistency of common finite volume methods for the Euler equations in the limit of low Mach number. In this work, the stationary states of the Discontinuous Galerkin (DG) method for linear acoustics on Cartesian grids are explored theoretically and experimentally, thus extending previous studies in the context of first-order finite difference methods. It is found that for a polynomial degree above some threshold, DG is stationarity preserving, but depending on the choice of numerical flux can suffer from a reduction of the order of accuracy at stationary state. This allows to explain the behaviour of the method for the Euler equations at low Mach number.
- J. Duan, W. Barsukow, C. Klingenberg:
An asymptotic-preserving active flux scheme for the hyperbolic heat equation in the diffusive scaling, 2025 submitted (pdf)- Abstract:The Active Flux (AF) method is a compact, high-order finite volume scheme that enhances flexibility by introducing point values at cell interfaces as additional degrees of freedom alongside cell averages. The method of lines is employed here for temporal discretization. A common approach for updating point values relies on the Jacobian Splitting (JS) method, which incorporates upwinding. A key advantage of the AF method over standard finite volume schemes is its structure-preserving property, motivating the investigation of its asymptotic-preserving (AP) behavior in the diffusive scaling. We show that the JS-based AF method without any modification is AP for solving the hyperbolic heat equation, in the sense that the limit scheme is a discretization of the limit heat equation. We use formal asymptotic analysis, discrete Fourier analysis, and numerical experiments to illustrate our findings.
- Abstract:The Active Flux (AF) method is a compact, high-order finite volume scheme that enhances flexibility by introducing point values at cell interfaces as additional degrees of freedom alongside cell averages. The method of lines is employed here for temporal discretization. A common approach for updating point values relies on the Jacobian Splitting (JS) method, which incorporates upwinding. A key advantage of the AF method over standard finite volume schemes is its structure-preserving property, motivating the investigation of its asymptotic-preserving (AP) behavior in the diffusive scaling. We show that the JS-based AF method without any modification is AP for solving the hyperbolic heat equation, in the sense that the limit scheme is a discretization of the limit heat equation. We use formal asymptotic analysis, discrete Fourier analysis, and numerical experiments to illustrate our findings.
Refereed journal articles
- W. Barsukow:
Semi-discrete Active Flux as a Petrov-Galerkin method: the case of one-dimensional and Cartesian grids, 2026 accepted in CAMC (pdf) - W. Barsukow:
An Active Flux method for the Euler equations based on the exact acoustic evolution operator, 2026 accepted in J. Sci. Comp. (pdf) - W. Barsukow, M. Ricchiuto, D. Torlo:
Stationarity preserving nodal Finite Element methods for multi-dimensional linear hyperbolic balance laws via a Global Flux quadrature formulation, 2026 accepted in J. Sci. Comp. (pdf) - W. Barsukow, C. Klingenberg, L. Lechner, J. Nordström, S. Ortleb, H. Ranocha:
Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators, 2026 accepted in BIT (pdf) - W. Barsukow, P. Chandrashekar, C. Klingenberg, L. Lechner:
A generalized Active Flux method of arbitrarily high order in two dimensions, Computers & Fluids (2025): 106886 (pdf, doi) - W. Barsukow, M. Ciallella, M. Ricchiuto, D. Torlo:
Genuinely multi-dimensional stationarity preserving global flux Finite Volume formulation for nonlinear hyperbolic PDEs, 2025 accepted (pdf) - A. Del Grosso, W. Barsukow, R. Loubère, P.-H. Maire:
An asymptotic-preserving multi-dimensional finite volume scheme for the Euler equations, Computers & Fluids (2025): 106951 (doi) - W. Barsukow, M. Ricchiuto, D. Torlo:
Structure preserving nodal continuous Finite Elements via Global Flux quadrature, Num. Meth. Part. Diff. Eq. (2025) 41(1): e23167 (pdf, doi) - Y. Liu, W. Barsukow:
An Arbitrarily High-Order Fully Well-balanced Hybrid Finite Element-Finite Volume Method for a One-dimensional Blood Flow Model, SISC (2025) 47(4): A2041-A2073 (pdf, doi) - W. Barsukow, J. Kern, C. Klingenberg, L. Lechner:
Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform, 2025 CAMC (2025): 1-49 (pdf, doi) - J. Duan, W. Barsukow, C. Klingenberg:
Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation, SISC (2025) 47(2): A811-A837 (pdf, combines 2405.02447 (1D) and 2407.13380 (multi-D), doi) - Remi Abgrall, Wasilij Barsukow, Christian Klingenberg:
A semi-discrete Active Flux method for the Euler equations on Cartesian grids, J. Sci. Comp. (2025) 102(2): 36 (pdf, hal, doi) - Wasilij Barsukow, Raphaël Loubère, Pierre-Henri Maire:
A node-conservative vorticity-preserving Finite Volume method for linear acoustics on unstructured grids, Math. Comp. (2025) 94(355): 2299-2343 (pdf, doi) - 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: Wed Jul 15 17:46:47 CET 2026