This paper is devoted to the analysis of the entropy stability properties of Active Flux-type scheme for a hyperbolic system equipped with one entropy inequality. This type of scheme evolves two sets of degrees of freedom: point values that are chosen on the boundary of the elements that cover the computational domain, and the average of the solution in these elements. We show that the only thing to do is to get an entropy inequality for the average values, the point values degrees of freedom do not play any role. We construct a monolithic scheme which is bound preserving of , non oscillatory following , and entropy diminishing. The entropy condition is implemented in Tadmor's framework, i.e. for the semi-discrete scheme only. The scheme is tested on the Kurganov-Popov-Petrova test case which is known to be sensitive to the satisfaction of an entropy inequality. We show that our entropy correction is effective: if we do not activate the bound-preserving nor the non oscillatory condition, we get the correct solution with some spurious wiggles, as expected. Though the development, implementation and tests are done with the triangle version of the scheme, the same method can be used for polygonal meshes, following .

We present a new class of structure-preserving semi-discrete continuous-discontinuous Galerkin (CG-DG) finite element schemes for linear and nonlinear hyperbolic systems of partial differential equations on unstructured simplex meshes that automatically satisfy the following properties: i) the new schemes are not only cellwise conservative, but also locally pointwise conservative everywhere, hence they satisfy the integral form of the conservation law on arbitrary control volumes that do not have to coincide with the mesh at all; ii) the new methods naturally satisfy the two basic vector calculus identities ∇·∇×𝐀 and ∇×∇ Z exactly pointwise locally and globally everywhere on the discrete level; iii) for linear symmetric hyperbolic systems the schemes are naturally energy conservative for the square energy, i.e. nonlinearly stable in the L^2 norm. The key ingredient of the new CG-DG schemes is the use of two different but compatible approximation spaces: the classical DG space 𝒰_h^N of discontinuous piecewise polynomials of degree up to N and a classical finite element space 𝒲_h^N+1 of globally continuous piecewise polynomials of degree N+1. In the new CG-DG schemes, the discrete solution 𝐮_h is sought in 𝒰_h^N, while a suitable discrete flux field 𝐟̃_h is computed in 𝒲_h^N+1. For N=0 our new schemes are directly related to cell-centered finite volume schemes with suitable vertex-based fluxes. All claimed properties of the schemes are first mathematically proven and are then also verified via suitable numerical tests. We show applications of our approach to three linear and nonlinear hyperbolic systems.

In this paper, we provide a few new properties of Active Flux (AF)/Point-Average-Moment PolynomiAl-interpreted (PamPa) schemes. First, we show, in full generality, that the AF/PamPa schemes can be interpreted in such a way that the discontinuous Galerkin (dG) scheme is one of their building blocks. Secondly we provide intrinsic bound preserving properties of the current variant of PamPa. This is also illustrated numerically. Last, we show, at least in one dimension, that the PamPa scheme has the summation by part (SBP) property.

We propose an improved version of the PAMPA algorithm where the solution is sought as globally continuous. The scheme is locally conservative, and there is no mass matrix to invert. This method had been developed in a series of papers, see e.g [1] and the references therein. In [2], we had shown the connection between PAMPA and the discontinuous Galerkin method, for the linear hyperbolic problem. Taking advantage of this reinterpretation, we use it to define a family of methods, show how to implement the boundary conditions in a rigorous manner. In addition, we propose a method that complements the bound preserving method developed in [2] in the sense that it is non oscillatory. A truncation error analysis is provided, it shows that the scheme should be third order accurate for smooth solutions. This is confirmed by numerical experiments. Several numerical examples are presented to show that the scheme is indeed bound preserving and non oscillatory on a wide range on numerical benchmarks.

In this work, we introduce new second-order schemes for one- and two-dimensional hyperbolic systems of conservation laws. Following an approach recently proposed in [R. Abgrall, Commun. Appl. Math. Comput., 5 (2023), pp. 370–402], we consider two different formulations of the studied system (the original conservative formulation and a primitive one containing nonconservative products), and discretize them on overlapping staggered meshes using two different numerical schemes. The novelty of our approach is twofold. First, we introduce an original paradigm making use of overlapping finite-volume (FV) meshes over which cell averages of conservative and primitive variables are evolved using semi-discrete FV methods: The nonconservative system is discretized by a path-conservative central-upwind scheme, and its solution is used to evaluate very simple numerical fluxes for the discretization of the original conservative system. Second, to ensure the nonlinear stability of the resulting method, we design a post-processing, which also guarantees a conservative coupling between the two sets of variables. We test the proposed semi-discrete dual formulation finite-volume methods on several benchmarks for the Euler equations of gas dynamics.

We propose a bound-preserving (BP) Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme by blending third-order and first-order constructions. The originality of the present construction is that it does not need any explicit reconstruction within each element, and therefore the construction is very flexible. The scheme employs a classical blending approach between a first-order BP scheme and a high-order scheme that does not inherently preserve bounds. The proposed BP PAMPA scheme demonstrates effectiveness across a range of problems, from scalar cases to systems such as the Euler equations of gas dynamics. We derive optimal blending parameters for both scalar and system cases, with the latter based on the recent geometric quasi-linearization (GQL) framework of [Wu & Shu, SIAM Review, 65 (2023), pp. 1031–1073]. This yields explicit, optimal blending coefficients that ensure positivity and control spurious oscillations in both point values and cell averages. This framework incorporates a convex blending of fluxes and residuals from both high-order and first-order updates, facilitating a rigorous BP property analysis. Sufficient conditions for the BP property are established, ensuring robustness while preserving high-order accuracy. Numerical tests confirm the effectiveness of the BP PAMPA scheme on several challenging problems.

The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method, proposed in [1], is a compact active-flux-type framework that combines conservative and nonconservative formulations of hyperbolic problems. In this paper, we develop a novel positivity-preserving (PP) PAMPA scheme for the ideal magnetohydrodynamics (MHD) equations on Cartesian grids. The method enforces a point-value-level discrete divergence-free (DDF) constraint on the intermediate interface states used in the update procedure, while the fully evolved continuous quadratic representation is not claimed to be exactly divergence-free at every instant. Building on our recent one-dimensional invariant-domain-preserving PAMPA framework [2], we extend the methodology to the multidimensional MHD system. The proposed scheme features an automatic PP update of interface point values through a new nonconservative reformulation, together with a local DDF projection. The cell-average update is provably PP under a mild a priori positivity condition on a single cell-centered value and combines four ingredients: (i) a DDF constraint at interface point values, (ii) a PP limiter applied only to the cell-centered value, (iii) a PP numerical flux with properly estimated wave speeds, and (iv) a suitable discretization of the Godunov–Powell source term. The PP proof for cell averages is carried out within the geometric quasi-linearization (GQL) framework [3], which transforms the nonlinear pressure-positivity constraint into an equivalent linear form. The resulting scheme avoids explicit polynomial reconstruction in practice, is compatible with arbitrarily high-order strong-stability-preserving time discretizations, and is straightforward to implement. To enhance robustness and resolution, we introduce a problem-independent troubled-cell indicator based on a Lax-type entropy criterion that uses only two characteristic speeds computed from cell averages and preserves symmetry, together with a convex oscillation suppression (COS) mechanism with a new distance measuring solution differences between target and neighboring cells. Extensive tests, including a rotated shock tube, a blast wave with plasma beta as low as 2.51×10−6, and a jet with Mach number up to 104, demonstrate high-order accuracy, sharp resolution of complex MHD structures, strong-shock robustness, and bounded analytical/discrete divergence diagnostics when these are evaluated from the unlimited representation. To the best of our knowledge, this is the first active-flux-type method for ideal MHD that combines a rigorous PP proof for cell averages, automatic admissibility of interface point values, and a point-value-level DDF mechanism in the update procedure.

The purpose of this paper is to discuss the notion of conservation in hyperbolic systems and how one can formulate it at the discrete level depending on the representation of the solution on the mesh. Since it is impossible to have a fully general theory, we discuss several alternatives possibilities: cases where the solution is represented by average in volumes; cases where the mesh is staggerred (i.e. the components of the solution are not localized at the same places); cases where the solution is solely represented by point values; and an example where all the previous options are mixed. We show how each configuration can provide, or not, enough flexibility. Though the discussion could be adapted to any hyperbolic system endowed with an entropy, we focus on compressible fluid mechanics, it its Eulerian and Lagrangian formulations. On a given mesh, the unifying element is that we systematically express the update of conserved variables as un+1=un-Delta t delta u, where the functional u delta u depends on the value of u at the current degree of freedom and its values from a set of degrees of freedom. This set defines the stencil of the scheme. From the stencil, one can naturally define a graph connecting the states that appears in delta u. The notion of local conservation can be defined from this graph. We are aware of only two possible situations: either the graph is constructed from the faces of the mesh elements (or the dual mesh), or it is defined from the mesh itself. Two notions of local conservation then emerge: either we define a numerical flux, or we define a "residual" attached to elements and the degrees of freedom within the element. We show that this two notions are in a way equivalent, but the one with residual allows much more flexibility, especially if additional algebraic constraints must be satisfied. Examples of specific additional conservation constraints are provided to illustrate this flexibility. We also show that this notion of conservation gives a very clear framework for the design of schemes in the Lagrangian setting. In the ending section, we will provide a number of ongoing research avenues strongly related to the formulation discussed, and we highlight some open questions which will be explored in the future.

In this paper, we explore the use of the Virtual Element Method (VEM) concepts to solve scalar and system hyperbolic problems on general polygonal grids. The new schemes stem from the Active Flux approach [1], which combines the usage of point values at the element boundaries with an additional degree of freedom representing the average of the solution within each control volume. Along the lines of the family of residual distribution schemes introduced in [2, 3] that integrate the Active Flux technique, we devise novel third order accurate methods that rely on the VEM technology to discretize gradients of the numerical solution by means of a polynomial-free approximation, by adopting a virtual basis that is locally defined for each element. The obtained discretization is globally continuous, and for nonlinear problems it needs a stabilization which is provided by a monolithic convex limiting strategy extended from [4]. This is applied to both point and average values of the discrete solution. We show applications to scalar problems, as well as to the acoustics and Euler equations in two dimension. The accuracy and the robustness of the proposed schemes are assessed against a suite of benchmarks involving smooth solutions, shock waves and other discontinuities.

We present a novel hydrostatic and non-hydrostatic equilibria preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) method for solving the one-dimensional hyperbolic balance laws, with applications to the shallow water models including the Saint--Venant system with the Manning friction term and rotating shallow water equations. The idea is based on a global flux quadrature formulation, in which the discretization of the source terms is obtained from the derivative of and additional flux function computed via high order quadrature of the source term. The reformulated system is quasi-conservative with global integral terms computed using Gauss--Lobatto quadrature nodes. The resulting method is capable of preserving a large family of smooth moving equilibria: supercritical and subcritical flows, in a super-convergent manner. We also show that, by an appropriate quadrature strategy for the source, we can exactly preserve the still water states. Moreover, to guarantee the positivity of water depth and eliminate the spurious oscillations near shocks, we blend the high-order PAMPA schemes with the first order local Lax--Friedrichs schemes using the method developed in [R. Abgrall, M. Jiao, Y. Liu, and K. Wu, arXiv preprint arXiv:2410.14292, 2024]. The first-order schemes are designed to preserve the still water equilibria and positivity of water height, as well as to deal with wet-dry fronts. Extensive numerical experiments are tested to validate the advantages and robustness of the proposed scheme.

We present a novel positive kinetic scheme built on the efficient collide-and-stream algorithm of the lattice Boltzmann method (LBM) to address hyperbolic conservation laws. We focus on the compressible Euler equations with strong discontinuities. Starting from the work of Jin and Xin [20] and then [4,8], we show how the LBM discretization procedure can yield both first- and second-order schemes, referred to as vectorial LBM. Noticing that the first-order scheme is convex preserving under a specific CFL constraint, we develop a blending strategy that preserves both the conservation and simplicity of the algorithm. This approach employs convex limiters, carefully designed to ensure either positivity (of the density and the internal energy) preservation (PP) or well-defined local maximum principles (LMP), while minimizing numerical dissipation. On challenging test cases involving strong discontinuities and near-vacuum regions, we demonstrate the scheme accuracy, robustness, and ability to capture sharp discontinuities with minimal numerical oscillations.

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 et al., ESAIM: Mathematical Modelling and Numerical Analysis 57:991-1027, 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.

The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method, proposed in [R. Abgrall, Commun. Appl. Math. Comput., 5: 370-402, 2023], combines conservative and non-conservative formulations of hyperbolic conservation laws to evolve cell averages and point values. Solutions to hyperbolic conservation laws typically have an invariant domain, and ensuring numerical solutions stay within this domain is essential yet nontrivial. This paper presents a novel framework for designing efficient Invariant-Domain-Preserving (IDP) PAMPA schemes. We first analyze the IDP property for updated cell averages in the original PAMPA scheme, revealing the role of cell average decomposition and midpoint values in maintaining the invariant domain. This analysis highlights the difficulty of relying on continuous fluxes alone to preserve the invariant domain. Building on these insights, we introduce a simple IDP limiter for cell midpoint values, and propose a provably IDP PAMPA scheme that guarantees the preservation of the invariant domain for updated cell averages without requiring post-processing limiters. This approach contrasts with existing bound-preserving PAMPA schemes, which often require additional convex limiting to blend high-order and low-order solutions. Most notably, inspired by the Softplus and Clipped ReLU functions from machine learning, we propose an automatic IDP reformulation of the governing equations, resulting in an unconditionally limiter-free IDP scheme for evolving point values. We also introduce techniques to suppress spurious oscillations, enabling the scheme to capture strong shocks effectively. Numerical experiments on 1D problems, including the linear convection equation, Burgers equation, the compressible Euler equations, and MHD equations, demonstrate the accuracy and robustness of the proposed IDP PAMPA scheme.

In this paper, we present an extension to non-uniform meshes of a 1D scheme [R & eacute;mi Abgrall and Davide Torlo. "Some preliminary results on a high order asymptotic preserving computationally explicit kinetic scheme". In: Abgrall and Torlo (2022). This scheme is arbitrary high order convergent in space and time for any hyperbolic system of conservation laws. It is based on a Finite Difference technique. We show that this numerical method is not conservative but it satisfies a Lax-Wendroff theorem under restrictive conditions on the mesh. To relax this condition we propose a Finite Volume alternative. This new discretization can be seen as a direct generalization to non-uniform meshes of the Finite Difference schemes in the sense that the fluxes of both methods are the same on uniform meshes. We apply the two schemes to the Euler system and we assess their performances on regarding test problems of the literature.

We develop a positivity-preserving (PP) PAMPA (Point-Average-Moment PolynomiAl-interpreted) scheme that enforces a discrete divergence-free (DDF) magnetic field for ideal MHD on Cartesian grids. Extending our 1D invariant-domain-preserving (IDP) PAMPA framework (Abgrall, Jiao, Liu, Wu, SIAM J. Sci. Comput., to appear) to multidimensional, multiwave MHD, the method combines a limiter-free PP update of interface point values via a new nonconservative reformulation with a local DDF projection. Cell averages are provably PP under a mild a~priori positivity condition on one cell-centered state, using: (i) DDF-constrained interface values, (ii) a PP limiter only at the cell center, (iii) a PP flux with appropriate wave-speed bounds, and (iv) a suitable discretization of the Godunov--Powell source term. The PP proof employs geometric quasi-linearization (GQL; Wu & Shu, SIAM Review, 2023), which linearizes the pressure constraint. The scheme avoids explicit polynomial reconstructions, is compatible with arbitrarily high-order strong-stability-preserving (SSP) time integration, and is simple to implement. Robustness and resolution are enhanced by a problem-independent Lax-type entropy troubled-cell indicator using only two characteristic speeds and a convex oscillation elimination (COE) mechanism with a new intercell-difference norm. Tests -- including a blast wave with plasma $β\approx 2.51\times 10^{-6}$ and jets up to Mach $10^{4}$ -- show high-order accuracy, sharp MHD-structure resolution, and strong-shock robustness. To our knowledge, this is the first active-flux-type ideal-MHD method rigorously PP for both cell averages and interface point values while maintaining DDF throughout.

In this paper we present a temperature update scheme that conserves the total energy and is valid for an arbitrary equation of state and spatial discretization. We derive the scheme for the single-phase Euler equations and present results for a shock tube spanning from super-critical to near-saturation conditions, measuring total energy imbalance and computational costs associated with thermodynamic evaluations. We also showcase non-classical results obtained with a multi-phase extension of the scheme for the full disequilibrium Baer-Nunziato equations.

In this work, we present a primitive update scheme for the full-disequilibrium Baer-Nunziato equations that is conservative in the total energy of the mixture and valid for generic equations of state. The update scheme is derived for a generic thermodynamic variable and is independent of the chosen spatial discretization. We show results of various Riemann problems from the literature obtained by updating phasic temperatures through the proposed scheme and compare them to the standard approach and analytical solutions. The total energy imbalance of the mixture is assessed, and computational speed-ups using the Span-Wagner equation of state are briefly discussed. Finally, the scheme is tested in complex thermodynamic conditions on a two-phase non-ideal and a two-fluid non-classical Riemann problem, using the Span-Wagner equation of state with vanishing phases.

We present a novel discretisation strategy, strongly inspired from Roe's Active Flux scheme. It can use polygonal meshes and is provably bound preserving for scalar problems and the Euler equations. Several cases demonstrates the quality of the method, and improvements with respect to previous work of the authors. This paper is a summary of \cite{BPPampa}.