Weekly Seminar

Abstract

We present kinetic type methods able to approximate compressible type flow, with or without viscous and thermal effects. Many numerical example illustrate the methods and show effectiveness. The work is strongly inspired from [1,2,3,4,5].

[1] R. Abgrall and D. Torlo. Some preliminary results on a high order asymptotic preserving computationally explicit kinetic scheme. Communications in Mathematical Sciences, 20(2):297–326, 2022.
[2] R. Abgrall and F. Nassajian Mojarrad. An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic. Communications on Applied Mathematics and Computation, (0123456789), 2023.
[3] Gauthier Wissocq and Rémi Abgrall. A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds. I: One-dimensional case. J. Comput. Phys., 518:29, 2024. Id/No 113333.
[4] Gauthier Wissocq and Rémi Abgrall. A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds. II: Multi-dimensional case. J. Comput. Phys., 516:27, 2024. Id/No 113376.
[5] Gauthier Wissocq, Yongle Liu, and Rémi Abgrall. A positive- and bound-preserving vectorial lattice Boltzmann method in two dimensions. SIAM SISC, accepted, 2025. ArXiv 2411.15001.  

Abstract

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 [4], 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 [1, 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 [2]. 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.

This is a joint work with W. Boscheri (Laboratoire de Mathematiques UMR 5127 CNRS, Université Savoie Mont Blanc, France), and Yongle Liu (Institute of Mathematics, University of Zurich, Switzerland)

[1] R. Abgrall. A combination of residual distribution and the active flux formulations or a new class of schemes that can combine several writings of the same hyperbolic problem: application to the 1d Euler equations. Commun. Appl. Math. Comput., 5(1):370–402, 2023.
[2] R. Abgrall, M. Jiao, Y. Liu, and K. Wu. Bound preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme: one-dimensional case. submitted, 2024. Arxiv: 2410.14292.
[3] R. Abgrall, J. Lin, and Y. Liu. Active flux for triangular meshes for compressible flows problems. Beijing Journal of Pure and Applied Mathematics, 2025. in press, also Arxiv preprint 2312.11271.
[4] T.A. Eyman and P.L. Roe. Active flux. 49th AIAA Aerospace Science Meeting, 2011.

Abstract

Hyperbolic balance laws can have particular moving equilibria involving particular source terms or based on divergence-free structures. Maintaining at a discrete level such equilibria (either exactly or very accurately) can give strong advantages with respect to classical methods. The Global Flux technique has been introduced to preserve 1D equilibria with source integrating the source into the flux term and has shown an increased accuracy in combination with many discretizations (FV, FD, FEM, DG). The 2D extensions have shown promising results in the linear acoustics case for vortex-like solutions. We will see what are the limits extending this method to nonlinear and multidimensional problems.

Abstract

Stochastic Galerkin formulations to random hyperbolic equations are based on the idea that the functional dependence on the stochastic input is described a priori by a polynomial chaos expansion and a Galerkin projection is used to obtain deterministic evolution equations for the coefficients in the series. Then, all envolved mathematical operations, e.g. products and norms, must be adopted and applied to the variables in the governing equations. In general, results for hyperbolic systems are not available, since desired properties like hyperbolicity and the existence of entropies are not transferred to the intrusive formulation. To this end, auxiliary variables have been introduced to establish wellposedness results. For instance, entropy-entropy flux pairs can be obtained by an expansion in entropy variables, i.e. the gradient of the deterministic entropy. Roe variables, which include the square root of the density, preserve hyperbolicity for Euler equations. The drawback of introducing auxiliary variables is an additional computational overhead that arises from an optimization problem, which is required to calculate the auxiliary variables. These results exploit quadratic relationships that are expressed efficiently by the Galerkin product. Extensions of the classical polynomial chaos expansions to general nonlinearities, e.g. with Legendre or Hermite polynomials, however, are not straightforward. More general nonlinearities, which occur for instance in isentropic Euler, level-set equations or when including source terms in shallow water equations, can be expressed by wavelet families that are generated by Haar-type matrices. Those are widely used in signal processing, e.g. for the Hadamard, Walsh, Chebyshev matrices and for the discrete cosine transform. In this talk we give an overview on methods to circumvent the loss of hyperbolicity and discuss briefly numerical challenges.

Abstract

Originally developed to model reaction-diffusion systems, Gradient Random Walk methods are particle-based techniques that track the evolution of spatial derivatives of a solution. Inspired by vortex methods for the Navier-Stokes equations, these approaches emerged in the 1990s and attracted significant interest due to their advantageous properties: (i) they eliminate the need for a computational grid, (ii) they naturally adapt to solution features by concentrating particles where gradients are steep, and (iii) they offer a notable reduction in variance.

In this seminar, we revisit and extend these ideas by demonstrating how the Gradient Random Walk framework can be adapted to a wider class of partial differential equations. To achieve this goal, we first extend the classical Monte Carlo method to the relaxation approximation of systems of conservation laws, and subsequently introduce a novel particle dynamics approach based on the spatial derivatives of the solution. This methodology, when combined with an asymptotic-preserving splitting discretization, enables the development of a new class of gradient-based Monte Carlo methods tailored for hyperbolic systems of conservation laws.

Abstract: The mathematical modeling of various real life phenomena involving the interplay of formation and consumption processes leads to non-linear Production--Destruction Systems (PDS) of the form y'(t) = (P(y(t)) - D(y(t))) e, where P,D:R^N \to \mathbb{R}^{N\times N}, \qquad \bm{e}=[1,\dots,1]^\mathsf{T}\in \mathbb{R}^N. In many practical applications, the underlying physics naturally yields PDS that are both positive and fully conservative for which $\Omega=\{ [x_1,\dots,x_N]^\mathsf{T} : 0 \leq x_i \leq \bm{e}^\mathsf{T}\bm{y}(t_0),  \ i=1,\dots,N \} $ is a positively-invariant set and the conservation law $\bm{e}^\mathsf{T}(\bm{y}(t)-\bm{y}(t_0))=0, \ \forall t\geq t_0$ holds true. As a result, there is an increasing demand for numerical integrators that retain both the positivity of the solution and the system's linear invariant with no limitations on the discretization steplength. In this context, \textit{Modified Patankar} (MP) methods have proven particularly beneficial (see \cite{Buchard,Huang2019,Izgin2024,Offner} and references therein). The purpose of this contribution is twofold. First, we present the results of \cite{Izzo2025}, where a novel class of $k$-step, unconditionally positive and conservative \textit{Modified Patankar Linear Multistep} (MPLM) methods is introduced. A thorough theoretical analysis is conducted, with emphasis placed upon the conditions on the \textit{Patankar Weight Denominators} (PWDs) required to achieve arbitrarily high orders of convergence. Additionally, the \(\sigma\)-embedding technique is proposed as a practical tool for the effective computation of PWDs. A comparison with other well-established MP discretizations reveals the competitiveness of the proposed methods for the simulation of PDS. \\ The second part of the present work explores the findings of \cite{MPSL}, where the framework of a \textit{Controlled PDS} (CPDS) \bm{y}'(t) = \bigl(P(\bm{y}(t))\odot \mathcal{P}(\bm{\alpha}(t)) - D(\bm{y}(t))\odot \mathcal{D}(\bm{\alpha}(t))\bigr)\bm{e},\quad \bm{\alpha}: \mathbb{R}^+ \rightarrow A\subset\mathbb{R}^M, \; \qquad \; \mathcal{P},\mathcal{D}: A \rightarrow \mathbb{R}^{N\times N} is outlined with the aim of influencing the dynamics of a PDS without altering its inherent properties. A general finite horizon \textit{Optimal Control Problem} (OCP) is formulated and addressed via the dynamic programming approach. Specifically, the OCP is restated in terms of a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation, whose unique viscosity solution corresponds to the value function \cite{Crandall}. We then propose a parallel-in-space \textit{Modified Patankar Semi-Lagrangian} (MPSL) approximation scheme for the HJB equation \cite{Falcone} and design an uncondionally positive and conservative reconstruction procedure for the optimal control in feedback and open-loop forms. The application to two case studies, specifically enzyme catalyzed biochemical reactions and infectious diseases, highlights the advantages of the proposed methodology over classical semi-Lagrangian discretizations.