arXiv subjects

Wasilij Barsukow

Publications and source records attributed to Wasilij Barsukow.

At least 19 recordsLinked to original sources

arXiv2026

Semi-discrete Active Flux as a Petrov-Galerkin method: the case of one-dimensional and Cartesian grids

Active Flux (AF) is a numerical method for hyperbolic conservation laws, whose degrees of freedom are averages/moments and (shared) point values at cell interfaces. It has been noted previously in a heuristic fashion that it thus combines ideas from Finite Volume/Discontinuous Galerkin (DG) methods with a continuous approximation common in continuous Finite Element (CG) methods. This work shows that semi-discrete Active Flux methods can be obtained from a variational formulation through a particular choice of (biorthogonal) test functions. These latter being discontinuous, the new formulation emphasizes the intermediate nature of AF between DG and CG. Explicit constructions are given for the case of arbitrarily high-order Active Flux with additional moments in 1-d, and for the classical third-order Active Flux on two-dimensional Cartesian meshes.

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arXiv2026

An Active Flux method for the Euler equations based on the exact acoustic evolution operator

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.

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arXiv2026

Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators

The Active Flux method is a numerical method for conservation laws using a combination of cell averages and point values as independent degrees of freedom, based on ideas from finite volumes and finite differences. This unusual mix has been shown to work well in many situations. We expand the theoretical justifications of the Active Flux method by analyzing it from the point of view of summation-by-parts (SBP) operators, which are routinely used to analyze finite difference, finite volume, and finite element schemes. We investigate in what type of setting the Active Flux method can be formulated using classical or degenerate SBP operators, yielding a first and novel approach for showing the energy stability of the Active Flux method. We present the analysis for the one-dimensional scalar linear advection equation with periodic boundary conditions on a uniform grid.

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arXiv2026

On the equivalence of semi-discrete Active Flux and Discontinuous Galerkin methods and a comparison of their performance

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.

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arXiv2025

Genuinely multi-dimensional stationarity preserving Finite Volume formulation for nonlinear hyperbolic PDEs

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 of 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 stationarity preserving quadrature strategy that allows to naturally introduce genuinely multi-dimensional numerical fluxes. 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.

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arXiv2025

A generalized Active Flux method of arbitrarily high order in two dimensions

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.

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arXiv2025

Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids

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.

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arXiv2025

Stationarity preserving nodal Finite Element methods for multi-dimensional linear hyperbolic balance laws via a Global Flux quadrature formulation

We consider linear, hyperbolic systems of balance laws in several space dimensions. They possess non-trivial steady states, which result from the equilibrium between derivatives of the unknowns in different directions, and the sources. Standard numerical methods fail to account for this equilibrium, and include stabilization that destroys it. This manifests itself in a diffusion of states that are supposed to remain stationary. We derive new stabilized high-order Finite Element methods based on a Global Flux quadrature: we reformulate the entire spatial operator as a mixed derivative of a single quantity, referred to as global flux. All spatial derivatives and the sources are thus treated simultaneously, and our methods are stationarity preserving. Additionally, when this formulation is combined with interpolation on Gauss-Lobatto nodes, the new methods are super-convergent at steady state. Formal consistency estimates, and strategies to construct well-prepared initial data are provided. The numerical results confirm the theoretical predictions, and show the tremendous benefits of the new formulation.

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arXiv2025

An asymptotic-preserving active flux scheme for the hyperbolic heat equation in the diffusive scaling

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.

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arXiv2025

Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform

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 approxi\-mations 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.

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arXiv2025

Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation

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.

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arXiv2025

An Arbitrarily High-Order Fully Well-balanced Hybrid Finite Element-Finite Volume Method for a One-dimensional Blood Flow Model

We propose an arbitrarily high-order accurate, fully well-balanced numerical method for the one-dimensional blood flow model. The developed method employs a continuous solution representation, combining conservative and primitive formulations. Degrees of freedom are point values at cell interfaces and moments of conservative variables within cells. \bla{The well-balanced property -- ensuring exact preservation of zero and non-zero velocity steady-state solutions while accurately capturing small perturbations -- is achieved through two key components. First, in the evolution of the moments, a local reference steady-state solution is obtained and subtracted. Second, the point value update happens in equilibrium variables. Extensive numerical tests are conducted to validate the preservation of various steady-state solutions, robust capturing small perturbations to such solutions, and high-order accuracy for both smooth and discontinuous solutions.

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arXiv2024

A semi-discrete Active Flux method for the Euler equations on Cartesian grids

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. 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. We demonstrate that a new semi-discrete Active Flux method (first described in Abgrall&Barsukow, 2023 for one space dimension) can be used to solve nonlinear hyperbolic systems in multiple dimensions without requiring evolution operators. We focus here on the compressible Euler equations of inviscid hydrodynamics and third-order accuracy. We introduce a multi-dimensional limiting strategy and demonstrate the performance of the new method on both Riemann problems and subsonic flows.

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arXiv2024

Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation: One-dimensional case

The active flux (AF) method is a compact high-order finite volume method that evolves cell averages and point values at cell interfaces independently. Within the method of lines framework, the point value can be updated based on Jacobian splitting (JS), incorporating the upwind idea. However, such JS-based AF methods encounter transonic issues for nonlinear problems due to inaccurate upwind direction estimation. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to the transonic issue. To improve robustness, this paper also develops bound-preserving (BP) AF methods for one-dimensional 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 in high-order AF methods is rewritten as a convex combination of using 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. Several challenging benchmark tests are conducted to verify the accuracy, BP properties, and shock-capturing ability of the methods.

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arXiv2024

Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation: Two-dimensional case

This paper studies the active flux (AF) methods for two-dimensional hyperbolic conservation laws, focusing on the flux vector splitting (FVS) for the point value update and bound-preserving (BP) limitings, which is an extension of our previous work [J.M. Duan, W. Barsukow, C. Klingenberg, arXiv:2405.02447]. The FVS-based point value update is shown to address the mesh alignment issue that appeared in a quasi-2D Riemann problem along one axis direction on Cartesian meshes. Consequently, the AF methods based on the FVS outperform those using Jacobian splitting, which are prone to transonic and mesh alignment issues. A shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can reduce oscillations well. Some benchmark problems are tested to verify the accuracy, BP property, and shock-capturing ability of our BP AF method. Moreover, for the double Mach reflection and forward-facing step problems, the present AF method can capture comparable or better small-scale features compared to the third-order discontinuous Galerkin method with the TVB limiter on the same mesh resolution, while using fewer degrees of freedom, demonstrating the efficiency and potential of our BP AF method for high Mach number flows.

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arXiv2024

A node-conservative vorticity-preserving Finite Volume method for linear acoustics on unstructured grids

Instead of ensuring that fluxes across edges add up to zero, we split the edge in two halves and also associate different fluxes to each of its sides. This is possible due to non-standard Riemann solvers with free parameters. We then enforce conservation by making sure that the fluxes around a node sum up to zero, which fixes the value of the free parameter. We demonstrate that for linear acoustics one of the non-standard Riemann solvers leads to a vorticity preserving method on unstructured meshes.

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arXiv2024

Structure preserving nodal continuous Finite Elements via Global Flux quadrature

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.

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arXiv2023

Implicit Active Flux methods for linear advection

In this work we develop implicit Active Flux schemes for the scalar advection equation. At every cell interface we approximate the solution by a polynomial in time. This allows to evolve the point values using characteristics and to update the cell averages using fluxes obtained by integrating this polynomial. The resulting schemes have order of convergence up to five, but show only moderate oscillations with high frequencies for discontinuous solutions. In numerical experiments we compare the different methods and show an application to network flows.

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