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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}., Comment: Submitted for the proceedings of HYP 2024

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., Comment: 28 pages

In this paper, we explore the use of the Virtual Element Method concepts to solve scalar and system hyperbolic problems on general polygonal grids. The new schemes stem from the active flux approach \cite{AF1}, 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 \cite{Abgrall_AF,abgrall2023activefluxtriangularmeshes} to bridge 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, hence 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 the \textit{a posteriori} MOOD paradigm \cite{Mood1}. 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 2D. 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 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., Comment: 23 pages, 10 figures

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, {\em 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.

In this paper, we develop novel well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) numerical methods for solving the two-dimensional shallow water equations on triangular meshes. The proposed PAMPA methods use a globally continuous representation of the variables, with degree of freedoms (DoFs) consisting of point values on the edges and average values within each triangular element. The update of cell averages is carried out using a conservative form of the partial differential equations (PDEs), while the point values are updated using a non-conservative formulation. This non-conservative formulation can be expressed in terms of either conservative or primitive variables. This new class of schemes is proved to be well-balanced and positivity-preserving. We validate the performance of the proposed methods through a series of numerical experiments. The numerical results are as expected and confirm that the performance of the PAMPA method using primitive variables in the non-conservative formulation is comparable to that using only the conservative variables. This work represents a step forward in the development and application of PAMPA methods for solving hyperbolic balance laws.

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.

In this article, we show how to construct a numerical method for solving hyperbolic problems, whether linear or nonlinear, using a continuous representation of the variables and their mean value in each triangular element. This type of approach has already been introduced by Roe, and others, in the multidimensional framework under the name of Active flux, see \cite{AF1,AF2,AF3,AF4,AF5}. Here, the presentation is more general and follows \cite{Abgrall_AF,BarzukowAbgrall}. { Various} examples show the good behavior of the method in both linear and nonlinear cases, including non-convex problems. The expected order of precision is obtained in both the linear and nonlinear cases. This work represents a step towards the development of methods in the spirit of virtual finite elements for linear or nonlinear hyperbolic problems, including the case where the solution is not regular.

In this paper, we introduce a new approach for constructing robust well-balanced numerical methods for the one-dimensional Saint-Venant system with and without the Manning friction term. Following the idea presented in [R. Abgrall, Commun. Appl. Math. Comput. 5(2023), pp. 370-402], we first combine the conservative and non-conservative (primitive) formulations of the studied conservative hyperbolic system in a natural way. The solution is globally continuous and described by a combination of point values and average values. The point values and average values will then be evolved by two different forms of PDEs: a conservative version of the cell averages and a possibly non-conservative one for the points. We show how to deal with both the conservative and non-conservative forms of PDEs in a well-balanced manner. The developed schemes are capable of exactly preserving both the still-water and moving-water equilibria. Compared with existing well-balanced methods, this new class of scheme is nonlinear-equations-solver-free. This makes the developed schemes less computationally costly and easier to extend to other models. We demonstrate the behavior of the proposed new scheme on several challenging examples.

One of the open problems in the field of forward uncertainty quantification (UQ) is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs. Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems, particularly with high dimensional random parameters. We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in (Ahlfeld et al. 2016) to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approximation via $\ell^1$ minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned $\ell^1$ minimization problem, we sample from the so-called induced distribution, instead of using Monte Carlo (MC) sampling from the original, unknown probability measure. We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures (such as the equilibrium measure) when we have incomplete information about the distribution. We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus., Comment: 16 pages

We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients---the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria. The designed scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments.

In this work, we combine the idea of data-driven polynomial chaos expansions with the weighted least-square approach to solve uncertainty quantification (UQ) problems. The idea of data-driven polynomial chaos is to use statistical moments of the input random variables to develop an arbitrary polynomial chaos expansion, and then use such data-driven bases to perform UQ computations. Here we adopt the bases construction procedure by following \cite{Ahlfeld_2016SAMBA}, where the bases are computed by using matrix operations on the Hankel matrix of moments. Different from previous works, in the postprocessing part, we propose a weighted least-squares approach to solve UQ problems. This approach includes a sampling strategy and a least-squares solver. The main features of our approach are two folds: On one hand, our sampling strategy is independent of the random input. More precisely, we propose to sampling with the equilibrium measure, and this measure is also independent of the data-driven bases. Thus, this procedure can be done in prior (or in a off-line manner). On the other hand, we propose to solve a Christoffel function weighted least-square problem, and this strategy is quasi-linearly stable -- the required number of PDE solvers depends linearly (up to a logarithmic factor) on the number of (data-driven) bases. This new approach is thus promising in dealing with a class of problems with epistemic uncertainties. Several numerical tests are presented to show the effectiveness of our approach., Comment: 20 pages, 10 figures

We study the properties of sparse reconstruction of transformed $\ell_1$ (TL1) minimization and present improved theoretical results about the recoverability and the accuracy of this reconstruction from undersampled measurements. We then combine this method with the stochastic collocation approach to identify the coefficients of sparse orthogonal polynomial expansions for uncertainty quantification. In order to implement the TL1 minimization, we use the DCA-TL1 algorithm which was introduced by Zhang and Xin. In particular, when recover non-sparse functions, we adopt an adaptive DCA-TL1 method to guarantee the sparest solutions. Various numerical examples, including sparse polynomial functions recovery and non-sparse analytical functions recovery are presented to demonstrate the recoverability and efficiency of this novel method and its potential for problems of practical interests., Comment: 18 pages, 8 figures