Publications
Lists of publications on preprint servers: arXiv, hal
Preprints
- W. Barsukow, M. Ciallella, M. Ricchiuto, D. Torlo:
Genuinely multi-dimensional stationarity preserving global flux Finite Volume formulation for nonlinear hyperbolic PDEs, 2025 submitted- Abstract:Classical Finite Volume methods for multi-dimensional problems include stabilization (e.g.\ via a Riemann solver), that is derived by considering several one-dimensional problems in different directions. Such methods therefore ignore a possibly existing balance of contributions coming from different directions, such as the one characterizing multi-dimensional stationary states. Instead being preserved, they are usually diffused away by such methods. Stationarity preserving methods use a better suited stabilization term that vanishes at the stationary state, allowing the method to preserve it. This work presents a general approach to stationarity preserving Finite Volume methods for nonlinear conservation/balance laws. It is based on a multi-dimensional extension of the global flux approach.
The new methods are shown to significantly outperform existing ones even if the latter are of higher order of accuracy and even on non-stationary solutions.
- Abstract:Classical Finite Volume methods for multi-dimensional problems include stabilization (e.g.\ via a Riemann solver), that is derived by considering several one-dimensional problems in different directions. Such methods therefore ignore a possibly existing balance of contributions coming from different directions, such as the one characterizing multi-dimensional stationary states. Instead being preserved, they are usually diffused away by such methods. Stationarity preserving methods use a better suited stabilization term that vanishes at the stationary state, allowing the method to preserve it. This work presents a general approach to stationarity preserving Finite Volume methods for nonlinear conservation/balance laws. It is based on a multi-dimensional extension of the global flux approach.
- A. Del Grosso, W. Barsukow, R. Loubère, P.-H. Maire:
An asymptotic-preserving multi-dimensional finite volume scheme for the Euler equations, 2025 submitted- Abstract:In the context of the numerical approximation of Euler equations, great efforts have been devoted to developing schemes that can accurately reproduce solutions in low Mach number ows. Solutions of classic Finite Volume (FV) schemes are usually plagued by an excessive diffusion as the numerical scheme is not consistent with the limit equations for the Mach number that tends to zero. Instead, a numerical scheme that satisfies such a property is called Asymptotic-Preserving (AP). In this paper, we propose an AP FV scheme for the multidimensional Euler equations. Contrarily to classic multidimensional FV schemes, our method is not obtained by a dimensional splitting. Instead, we include genuine multidimensional effects by exploiting a particular Riemann Solver (RS). It involves a nodal parameter which depends on all the cells around a given node. Such an idea has been exploited by Barsukow et al. (2023) for the linear acoustic equations. Their method performs excellently in the low Mach number regime but its extension to the Euler equations proved to be far from trivial. For such a reason, a change of perspective is needed in the definition of the RS.
- Abstract:In the context of the numerical approximation of Euler equations, great efforts have been devoted to developing schemes that can accurately reproduce solutions in low Mach number ows. Solutions of classic Finite Volume (FV) schemes are usually plagued by an excessive diffusion as the numerical scheme is not consistent with the limit equations for the Mach number that tends to zero. Instead, a numerical scheme that satisfies such a property is called Asymptotic-Preserving (AP). In this paper, we propose an AP FV scheme for the multidimensional Euler equations. Contrarily to classic multidimensional FV schemes, our method is not obtained by a dimensional splitting. Instead, we include genuine multidimensional effects by exploiting a particular Riemann Solver (RS). It involves a nodal parameter which depends on all the cells around a given node. Such an idea has been exploited by Barsukow et al. (2023) for the linear acoustic equations. Their method performs excellently in the low Mach number regime but its extension to the Euler equations proved to be far from trivial. For such a reason, a change of perspective is needed in the definition of the RS.
- W. Barsukow:
An Active Flux method for the Euler equations based on the exact acoustic evolution operator, 2025 submitted (pdf)- Abstract:A new Active Flux method for the multi-dimensional Euler equations is based on an additive operator splitting into acoustics and advection. The acoustic operator is solved in a locally linearized manner by using the exact evolution operator. The nonlinear advection operator is solved at third order accuracy using a new approximate evolution operator. To simplify the splitting, the new method uses primitive variables for the point values and for the reconstruction. In order to handle discontinuous solutions, a blended bound preserving limiting is used, that combines a priori and a posteriori approaches. The resulting method is able to resolve multi-dimensional Riemann problems as well as low Mach number flow, and has a large domain of stability.
- Abstract:A new Active Flux method for the multi-dimensional Euler equations is based on an additive operator splitting into acoustics and advection. The acoustic operator is solved in a locally linearized manner by using the exact evolution operator. The nonlinear advection operator is solved at third order accuracy using a new approximate evolution operator. To simplify the splitting, the new method uses primitive variables for the point values and for the reconstruction. In order to handle discontinuous solutions, a blended bound preserving limiting is used, that combines a priori and a posteriori approaches. The resulting method is able to resolve multi-dimensional Riemann problems as well as low Mach number flow, and has a large domain of stability.
- 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.
Refereed journal articles
- W. Barsukow, M. Ricchiuto, D. Torlo:
Structure preserving nodal continuous Finite Elements via Global Flux quadrature, 2024 accepted (pdf) - W. Barsukow, Y. Liu:
An Arbitrarily High-Order Fully Well-balanced Hybrid Finite Element-Finite Volume Method for a One-dimensional Blood Flow Model, 2025 accepted (pdf) - W. Barsukow, J. Kern, C. Klingenberg, L. Lechner:
Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform, 2025 accepted in CAMC (pdf) - 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 Jun 23 20:25:40 CEST 2025