MC 5501
Speaker
Rémi Abgrall | Institut für Mathematik & Computational Science Universität Zürich
Title
Virtual finite element and hyperbolic problems: the PAMPA algorithm
Abstract
In this talk, 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 ux 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 ux 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 de ned 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.
References
[1] R. Abgrall. A combination of residual distribution and the active ux 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 ux for triangular meshes for compressible
ows 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 ux. 49th AIAA Aerospace Science Meeting, 2011.
Speaker Bio
Professor Rémi Abgrall has been a faculty member at the University of Zurich since 2014, and has served as Director of the Department of Mathematics since 2021. He is a renowned applied mathematician, recognized for his significant contributions to computational fluid dynamics, the numerical analysis of conservation laws, multiphase flows, and Hamilton–Jacobi equations.
In 2014, Professor Abgrall was an invited speaker at the International Congress of Mathematicians (ICM) held in Seoul. In recognition of his groundbreaking work on numerical methods for conservation laws—particularly in the areas of multi-fluid flows and residual distribution schemes—he was elected a SIAM Fellow in 2022.
In 2025, he received the Carl Friedrich von Siemens Research Award (also known as the Humboldt Research Award), the highest honor conferred by the Alexander von Humboldt Foundation. He was also awarded the 2024 ECCOMAS Prandtl Medal and previously received the GAMNI Prize from the French Academy of Sciences.
From 2015 to 2024, Professor Abgrall served as Editor-in-Chief of the Journal of Computational Physics.