Abstract:
Open session.

Chair: Kristof Cools and Le Van Chien (Ghent University)

Abstract:
The rapidly rising variety of acoustic and electromagnetic applications has demanded the high-precision computing capacity of wave equations in complicated scenarios, for instance, modelling and analyzing systems in the presence of wide bandwidth data, large-scale, multi-scale and multi-physics geometries with complex material properties. This demand has accelerated the critical need for studies of robust, stable and highly efficient computational methods, necessitating more and more efforts from both mathematicians, physicists, computer scientists, and engineers. Within this mini symposium, we aim to bring together international experts in the field of numerical and computational methods for acoustic and electromagnetic wave equations to discuss the latest research and knowledge, explore challenges and new directions, as well as to foster interaction and collaboration between communities. The topics of this mini symposium cover both frequency-domain and time-domain challenges, with a particular focus on finite and boundary element methods.

Chair: Minseok Choi (Pohang University of Science and Technology) and Jooyoung Hahn (Slovak University of Technology in Bratislava)

Abstract:
This symposium addresses recent developments and persistent challenges in using neural network solvers for partial differential equations (PDEs), with a particular focus on the limitations and potential of physics-informed neural networks (PINNs). Despite their innovative approach, PINNs often face significant hurdles such as poor convergence, dependency on large data sets, and difficulties in generalizing across different types of PDEs. The session will examine how integrating classical mathematical concepts and numerical methods has begun to overcome these issues, enhancing the robustness and accuracy of neural network approaches. The discussion will highlight the latest adaptations in neural networks that incorporate advanced sampling strategies, optimization of network architectures, and novel training techniques that reduce the computational burden. Additionally, the symposium will present specific applications where modified PINNs have shown improved performance in solving complex, nonlinear PDEs in real-time scenarios, and in sensor placement strategies that optimize data acquisition in sparse environments. By bringing together diverse perspectives from applied mathematics and engineering, the session aims to provide a comprehensive understanding of how traditional numerical methods can be seamlessly integrated with modern machine learning techniques to address some of the most challenging problems in scientific computing.

Chair: Anna Klünker (Leuphana University Lüneburg) and Vamika Rathi (Hamburg University of Technology)

Abstract:
Inertial particles exhibit distinct transport behaviors compared to passive or ideal fluid particles. Their study is critical for a wide range of applications, from industrial processes like chemical reactors to natural phenomena such as marine snow or the transport of micro- or macro plastics in the ocean. This mini-symposium will discuss recent advancements in the mathematical modelling, simulation, and analysis of inertial particle transport across various applications. Talks will present novel numerical approaches for solving the Maxey-Riley equation, a key model to describe the motion of small inertial particles. They will also present various Lagrangian analysis tools that have been developed in the last decades to investigate transport and mixing in fluid flows. These tools provide valuable insights into the dispersion patterns of inertial particles, offering a deeper understanding of their complex behaviors in fluid systems.

Chair: Jochen Schütz (Hasselt University) and Peter Frolkovic (Slovak University of Technology)

Abstract:
Traditionally, time integration for hyperbolic PDEs has been done using explicit time integration schemes. While these schemes are easier to analyze and implement, and very often yield highly efficient algorithms, they can suffer from severe stability restrictions when the underlying PDE becomes stiff due to, e.g., large wave speeds, large sources, geometries with high aspect ratio and the like. Recent years have hence seen much progress in the development and analysis of implicit time integration for hyperbolic (and related) PDEs. This minisymposium aims to bring together researchers that work on implicit, semi-implicit or implicit/explicit (IMEX) time integration for PDEs, with a particular focus on hyperbolic PDEs.

Chair: Barbara Kaltenbacher (University of Klagenfurt) and Vanja Nikolić (Radboud University)

Abstract:
The research on nonlinear acoustics has mainly been driven by numerous developing applications of ultrasound waves, including imaging, non-invasive therapy, and targeted drug delivery. In recent years, significant advances have been made in the mathematical aspects of nonlinear acoustics, and new, challenging open questions have been raised in the areas of accurate modeling, rigorous analysis, and efficient simulation. This minisymposium aims to bring together researchers working on this topic to encourage novel ideas, trends, and cooperation in this field.

Chair: D. Di Pietro (University of Montpellier), L. Beirao da Veiga (University of Milano-Bicocca), P. Antonietti (Politecnico di Milano) and J. Droniou (University of Montpellier)

Abstract:
Electromagnetic models display specific structures that challenge their mathematical analysis. The well-posedness of stationary versions (such as magnetostatics) depend on the topology of the domain - typically, the number of holes in it. Transient versions comprise two types of equations: evolution and constraints (divergence condition on the fields), the latter being preserved by the former. When considering charged fluids, electromagnetic and Navier-Stokes equations are coupled, giving rise to complex non-linear magnetohydrodynamics equations that also present certain structures of interest, such as the preservation of helicity. Other nonlinear equations also arise in the context of generalised electromagnetic models, such as the Einstein equations.

Most of these structures and complexities stem from the fact that electromagnetic equations are based on "incomplete" operators: the curl and the divergence. The properties and relations between these two operators, which explain the aforementioned structures, are embedded into the de Rham complex and its cohomology. Designing robust and accurate numerical methods for electromagnetic (and related) models is a complex task, and it requires to properly represent at the discrete level the underlying mathematical structures. This necessitates suitable discretisations of the divergence and curl, and the resulting schemes often lead to indefinite linear systems that present severe computational challenges.

This minisymposium will focus on numerical and computational aspects of models involving the curl and divergence operators, with emphasis on electromagnetic and magnetohydrodynamics models. Topics covered include the design and analysis of numerical methods for these models, considerations around the preservation of structures at the discrete level, computational challenges, etc.