Séminaire Calcul Scientifique et Modélisation

We discuss a new swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with position $x$ and mass $m$. There are three key aspects to the SBGD dynamics: (i) persistent transition of mass from agents at high to lower ground; (ii) a random marching direction, aligned with the steepest gradient descent; and (iii) a time stepping protocol which decreases with $m$.

The interplay between positions and masses leads to dynamic distinction between `heavier leaders’ near local minima, and `lighter explorers’ which explore for improved position with large(r) time steps. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer. 

The coupling of coastal wave models, such as Boussinesq-type (BT) and Saint-Venant (SV) equations, has been explored since the 1990s. Despite numerous models and coupling examples, the literature exhibits significant disagreement regarding induced artifacts and methods for their analysis. This work aims to elucidate these issues, proposing explanations and a method for evaluating and comparing coupling techniques. We ground our explanation in the mathematical properties of each model's Cauchy and half-line problems, highlighting the sensitivity of these models to numerical artifacts. Additionally, we demonstrate how one-way models provide insights into expected physical effects, unexpected artifacts, and errors relative to 3D models. We demonstrate this analysis with linearized models, where we establish the well-posedness of a popular coupling, characterize analytically the "coupling error" in terms of wave reflections, and prove its asymptotic behavior in shallow water. We will discuss how these insights can be applied to other linear/nonlinear models, providing a foundation for the evaluation and comparison of new coupled coastal wave models.

Simuler numériquement de manière précise l'évolution des interfaces séparant différents milieux est un enjeu crucial dans de nombreuses applications (multi-fluides, fluide-structure, etc). La méthode MOF (moment-of-fluid), extension de la méthode VOF (volume-of-fluid), utilise une reconstruction affine des interfaces par cellule basée sur les fractions volumiques et les centroïdes de chaque phase. Cette reconstruction d'interface est solution d'un problème de minimisation sous contrainte de volume. Ce problème est résolu dans la littérature par des calculs géométriques sur des polyèdres qui ont un coût important en 3D. On propose dans cet exposé une nouvelle approche du calcul de la fonction objectif et de ses dérivées de manière complètement analytique dans le cas de cellules hexaédriques rectangulaires et tétraédriques en 3D. Les résultats numériques montrent un gain important en temps de calcul.

We present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton’s principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler–Lagrange equation. A sufficient criterion for the hyperbolicity of the model is formulated. The governing equations are asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine–Hugoniot conditions and Clausius–Duhem inequality is performed for a specific choice of the equation of state. In particular, this reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.

This talk presents a family of algebraically constrained finite element schemes for hyperbolic conservation laws. The validity of generalized discrete maximum principles is enforced using monolithic convex limiting (MCL), a new flux correction procedure based on representation of spatial semi-discretizations in terms of admissible intermediate states. Semi-discrete entropy stability is enforced using a limiter-based fix. Time integration is performed using explicit or implicit Runge-Kutta methods, which can also be equipped with property-preserving flux limiters. In MCL schemes for nonlinear systems, problem-dependent inequality constraints are imposed on scalar functions of conserved variables to ensure physical and numerical admissibility of approximate solutions. After explaining the design philosophy behind our flux-corrected finite element approximations and showing some numerical examples, we turn to the analysis of consistency and convergence. For the Euler equations of gas dynamics, we prove weak convergence to a dissipative weak solution. The convergence analysis to be presented in this talk is joint work with Maria Lukáčová-Medvid’ová and Philipp Öffner.

Networks of hyperbolic PDEs arise in different applications, e.g. modeling water- or gas-networks or road traffic. In the first part of this talk we discuss modeling aspects of coupling conditions for hyperbolic PDEs.

Starting from an kinetic description we derive coupling conditions for the associated macroscopic equations. For this process a detailed description of the boundary layer is important. In the second part appropriate numerical methods are considered.

Different high order approaches are compared and applications to district heating or water networks are discussed.

Multidimensional simulations of magnetohydrodynamic phenomena occurring in stellar interiors are essential for understanding how stars evolve and die. The highly subsonic flow regimes found in the regions deep inside stars pose severe challenges to conventional methods of computational MHD, such as the popular "high-resolution shock-capturing'' schemes. After giving a brief overview of work on astrophysical simulations (including also supernova explosions and common-envelope evolution) in our group at Heidelberg, we summarize the challenges and present suitable numerical solvers optimized for magnetized, low-Mach-number stellar flows, implemented in our Seven-League Hydro code. We show how the choice of the numerical method can drastically affect both the performance of the code and its accuracy in real astrophysical simulations.

We study non-conservative hyperbolic systems of balance laws and are interested in development of well-balanced (WB) numerical methods for such systems. One of the ways to enforce the balance between the flux terms and source and non-conservative product terms is to rewrite the studied system in a quasi-conservative form by incorporating the latter terms into the modified global flux. The resulting system can be quite easily solved by Riemann-problem-solver-free central-upwind (CU) schemes. This approach, however, does not allow to accurately treat non-conservative products. We therefore apply a path-conservative (PC) integration technique and develop a very robust and accurate path-conservative central-upwind schemes (PCCU) based on flux globalization. I will demonstrate the performance of the WB PCCU schemes on a wide variety of examples.

This talk explores two advanced numerical methods for solving compressible two-phase flows modelled using the conservative Symmetric Hyperbolic Thermodynamically Compatible (SHTC) model proposed by Romenski et al. I first address the weak hyperbolicity of the original model in multidimensional cases by restoring strong hyperbolicity through two distinct approaches: the explicit symmetrization of the system and the hyperbolic Generalized Lagrangian Multiplier (GLM) curl-cleaning approach. Then, I will present two numerical methods to solve the proposed problem: a high-order ADER Discontinuous Galerkin (ADER-DG) scheme with an a posteriori sub-cell finite volume limiter and an exactly curl-free finite volume scheme to handle the curl involution in the relative velocity field. The latter method uses a staggered grid discretization and defines a proper compatible gradient and a curl operator to achieve a curl-free discrete solution. Extensive numerical test cases in one and multiple dimensions validate both methods' accuracy and stability. 

A Coulter counter is an impedance measurement system widely used in blood analyzers to count and size red blood cells, thus providing information about the most numerous cells of the body. In Coulter counters, cells flow through a detection zone where an electric field is imposed, which is disturbed when a cell passes through. The number of these impedance signals yield the red blood cell count, while the cell volume is supposed to be proportional to the amplitude of the signals. However, in real systems, the red blood cells trajectories in the system does not allow to verify the assumptions necessary to provide an accurate volume measurement. For a few years, IMAG has been developing the YALES2BIO solver for the prediction of red blood cell dynamics under flow. In this presentation, I will describe the fluid-structure problem and the numerical method used, then share how numerical simulation has been used to understand the signals in industrial Coulter counters and to improve the measurements of red blood cell volumes rendered by such systems. In addition, I will discuss how the mechanical properties of RBCs impact the measurements. This work has been performed during the PhD theses of Pierre Taraconat and Pierre Pottier (Horiba Medical & IMAG).

In this talk, we propose a novel collocation-based Model Order Reduction (cMOR) strategy for solving parametric advection-diffusion PDEs on moving Chimera grids. Unlike traditional projection-based MOR, cMOR solves the High-Dimensional Model on a small subset of collocation points and extends the solution to the entire domain using a global reduced basis. By leveraging the ADER method on unsteady Chimera meshes, cMOR addresses the computational challenges posed by convective dominated problems, particularly the Kolmogorov N-width barrier. Our results demonstrate the efficiency of cMOR in reducing the computational cost while maintaining accuracy.