École Thématique SPEGEO

Le groupe Calcul organise du 22 au 26 juin 2026 au Centre Culturel Saint-Thomas à Strasbourg une école thématique intitulée SPEGEO: Structure-Preserving Algorithms and Geometric Numerical Integration.

L'objectif de cette école est de présenter un panorama des méthodes d'intégration numérique préservant la structure géométrique associée aux équations de la dynamique d'un système physique. L'aspect mathématique de ces méthodes est abordé mais également des exemples de mise en œuvre pratique des algorithmes sont utilisés sous la forme de travaux pratiques. Elle est structurée autour de cinq cours-TP (3h+1h30):

slides GEMPICX code

Structure-preserving techniques enable qualitatively correct and robust numerical simulations. In this short course, we focus a geometrical particle discretization of the Vlasov-Maxwell system and introduce the various techniques used in the method.

The course be split in the following parts:

1. Structure-preserving methods for Maxwell's equation and the Finite Element Exterior Calculus Framework.
2. Discrete Lagrangian and Hamiltonian Principles.
3. Symplectic and energy-conserving time-discretizations.
4. Application to the non-relativistic and relativistic Vlasov-Maxwell models.
5. Hands-on exercises.

slides struphy code struphy-tutorials

In this course we explore the use of mesh-free methods for solving fluid dynamic equations. Such methods are also known as Smoothed-Particle-Hydrodynamics (SPH) [1, 2, 3], and are particularly suited for strongly deforming flows and complex geometries. The course is partitioned into three parts:

1. Theory - from Lagrangian to Eulerian coordinates via Euler-Poincaré reduction (90 min),
2. Particle discretizations of Lagrangians for isothermal Euler, polytropic Euler and incompressible Navier-Stokes equations (45 min),
3. Solving meshfree discretizations with Struphy: an open source Python package for scientific computing (135 min)

References:

• [1] J. J. Monaghan, "Smoothed particle hydrodynamics", Rep. Prog. Phys. 68 (2005) 1703–1759
• [2] V. Springel, "Smoothed Particle Hydrodynamics in Astrophysics", Annu. Rev. Astron. Astrophys. 2010. 48:391–430
• [3] D.J. Prince, "Smoothed particle hydrodynamics and magnetohydrodynamics", J. Comp. Phys. 231 (2012) 759–794

hands on

In this lecture, we will address someaspects linked to structure preservation in compressible fluid dynamics. We will start by considering the accuracy problem of usual density based upwind solvers in the low Mach number limit. After some numerical illustrations, we will develop the one scale nd two scales asymptotic expansion of the full Euler and barotropic Euler system in the low Mach number limit. Based on this asymptotic expansion, we will make a link between the accuracy problem at low Mach number, and the double limit (mesh convergence/long time behaviour) of the acoustic wave system. Being now interested in the long time limit of the acoustic wave system, we will first thoroughly study the special case of the wave system on triangular meshes. We will show that a special structure useful for studying the long time limit of the acoustic system, a Hodge-Helmholtz decomposition, exists also in the discrete case on triangular meshes. We will then show that this structure can be put in a distributional de-Rham complex. This general framework will allow to develop a similar theory for the high order discontinuous Galerkin method on triangular and quadrangular meshes. Based on these approximation spaces, we will show how to preserve a curl or a divergence constraint with the high order discontinuous Galerkin method. Examples will then include the Euler system at low Mach number, the magnetohydrodynamics system, the Maxwell system. We will also show some results on the preservation of nontrivial equilibrium states for some hyperbolic systems.

The Hybrid High-Order (HHO) method is a numerical method for the approximation of PDEs, that is of arbitrary order and applicable on generic polygonal/polyhedral meshes. HHO can be seen as an extension of the non-conforming P1 finite element to generic grids and polynomial degrees, and it can also be interpreted as a finite volume method. Using the Laplace equation as an illustration, we will show how an HHO scheme is constructed from local reconstruction operators and stabilisation terms (elements that are shared by many polytopal methods).

The implementation of the HHO scheme will also be covered, using a novel C++ DSEL (Domain-Specific Embedded Language) framework built on the HArDCore library. The DSEL allows for a seamless coding of the equations defining reconstruction operators in polytopal methods, with a language that mimics the mathematical formulas themselves.

• reference, and notes taken during the lectures.
• rough notes on the DSEL syntax, that you will have to use (with guidance!)