Publications
Lists of publications on preprint servers: arXiv, hal
Preprints
- Remi Abgrall, Wasilij Barsukow, Christian Klingenberg:
The Active Flux method for the Euler equations on Cartesian grids, 2023 submitted (pdf, hal)- Abstract: Active Flux is an extension of the Finite Volume method and additionally incorporates point values located at cell boundaries. This gives rise to a globally continuous approximation of the solution. The method is third-order accurate. We demonstrate that a new semi-discrete Active Flux method (first described in Abgrall&Barsukow, 2022 for one space dimension) can easily be used to solve nonlinear hyperbolic systems in multiple dimensions, such as the compressible Euler equations of inviscid hydrodynamics. Originally, the Active Flux method emerged as a fully discrete method, and required an exact or approximate evolution operator for the point value update. For nonlinear problems such an operator is often difficult to obtain, in particular for multiple spatial dimensions. With the new approach it becomes possible to leave behind these difficulties. We introduce a multi-dimensional limiting strategy and demonstrate the performance of the new method on both Riemann problems and subsonic flows.
- Abstract: Active Flux is an extension of the Finite Volume method and additionally incorporates point values located at cell boundaries. This gives rise to a globally continuous approximation of the solution. The method is third-order accurate. We demonstrate that a new semi-discrete Active Flux method (first described in Abgrall&Barsukow, 2022 for one space dimension) can easily be used to solve nonlinear hyperbolic systems in multiple dimensions, such as the compressible Euler equations of inviscid hydrodynamics. Originally, the Active Flux method emerged as a fully discrete method, and required an exact or approximate evolution operator for the point value update. For nonlinear problems such an operator is often difficult to obtain, in particular for multiple spatial dimensions. With the new approach it becomes possible to leave behind these difficulties. We introduce a multi-dimensional limiting strategy and demonstrate the performance of the new method on both Riemann problems and subsonic flows.
- J. Duan, W. Barsukow, C. Klingenberg:
Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation, 2024 submitted (pdf, combines 2405.02447 (1D) and 2407.13380 (multi-D))- Abstract:The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. Within the method of lines framework, the existing Jacobian splitting-based point value update incorporates the upwind idea but suffers from a stagnation issue for nonlinear problems due to inaccurate estimation of the upwind direction, and also from a mesh alignment issue partially resulting from decoupled point value updates. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to those two issues. To improve robustness, this paper also develops bound-preserving (BP) AF methods for hyperbolic conservation laws. Two cases are considered: preservation of the maximum principle for the scalar case, and preservation of positive density and pressure for the compressible Euler equations. The update of the cell average is rewritten as a convex combination of the original high-order fluxes and robust low-order (local Lax-Friedrichs or Rusanov) fluxes, and the desired bounds are enforced by choosing the right amount of low-order fluxes. A similar blending strategy is used for the point value update. In addition, a shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can suppress oscillations well. Several challenging tests are conducted to verify the robustness and effectiveness of the BP AF methods, including flow past a forward-facing step and high Mach number jets.
- Abstract:The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. Within the method of lines framework, the existing Jacobian splitting-based point value update incorporates the upwind idea but suffers from a stagnation issue for nonlinear problems due to inaccurate estimation of the upwind direction, and also from a mesh alignment issue partially resulting from decoupled point value updates. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to those two issues. To improve robustness, this paper also develops bound-preserving (BP) AF methods for hyperbolic conservation laws. Two cases are considered: preservation of the maximum principle for the scalar case, and preservation of positive density and pressure for the compressible Euler equations. The update of the cell average is rewritten as a convex combination of the original high-order fluxes and robust low-order (local Lax-Friedrichs or Rusanov) fluxes, and the desired bounds are enforced by choosing the right amount of low-order fluxes. A similar blending strategy is used for the point value update. In addition, a shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can suppress oscillations well. Several challenging tests are conducted to verify the robustness and effectiveness of the BP AF methods, including flow past a forward-facing step and high Mach number jets.
- 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.
Refereed journal articles
- Wasilij Barsukow, Raphaël Loubère, Pierre-Henri Maire:
A node-conservative vorticity-preserving Finite Volume method for linear acoustics on unstructured grids, 2024 accepted - 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 Nov 13 18:46:43 CET 2024