David Mora, Ricardo Ruiz-Baier

This MS aims to bring together researchers working on the theoretical analysis and applications of hybrid and polytopal methods. We will focuson novel discretizations for coupled partial differential equations (PDEs) arising in multiphysics models featuring for example interface coupled systems, also welcoming contributions addressing the approximation of eigenvalue problems. The MS will provide an opportunity for researchers to share their knowledge of developing efficient numerical methods and algorithms for solving these challenging problems.

Bao Weizhu, Yue Feng

Nonlinear partial differential equations (PDEs) play a fundamental role in many fields from science to engineering. Nonlocal modeling is quite popular in describing some physical phenomena involving long-range interactions. The mathematical models and solutions of PDEs with low regularity arise in some practical problems. From the perspective of computational mathematics, it is significant to design high-order methods to solve PDEs with low regularity and nonlocal terms and provide an intuitive view for physical phenomena. The proposed minisymposium brings together experts from computational mathematics to provide an overview of current state-of-the-art and recent advances in the design and analysis of high-order methods for nonlinear PDEs with low regularity and nonlocal terms as well as their applications in various fields.

Moritz Hauck, Zhi-Song Liu, Andreas Rupp,

This mini-symposium explores the dynamic convergence of machine learning and numerical methods, highlighting their potential to forge innovative and powerful computational tools. As machine learning continues to revolutionize various disciplines, its integration with traditional numerical techniques promises to be transformative, particularly in areas such as inverse problems, partial differential equations, sampling, and classification. The symposium will feature insights from leading experts and emerging researchers, who will share their experiences and highlight successful collaborations between these two fields. The goals of the mini-symposium include: (i) to explore the synergies between machine learning and numerical methods, (ii) foster interdisciplinary collaborations through engaging discussions, and (iii) uncover novel approaches to address complex computational challenges. This event aims to be a platform for inspiring ideas, facilitating networking, and driving future advancements at the intersection of these two exciting fields.

Emma Perracchione, Elisabeth Larsson

Kernels, and in particular Radial Basis Functions (RBFs), have gained increasing popularity as a meshfree methodology. Originally proposed by R. Hardy in 1971 for interpolating scattered data, RBF-based methods have evolved significantly, and they have been successful in various fields of numerical analysis, including scientific machine learning and approximation of PDEs. Their widespread use is largely due to their ability to achieve spectral convergence on arbitrarily distributed nodes. However, one of the main challenges associated with kernel methods is that the resulting collocation matrices are dense or only moderately sparse, and often ill-conditioned. Hence, many efforts have been dedicated to propose more stable and computationally efficient methods and algorithms able to deal with large amounts of data (see e.g. Contour-Pad, RBF-QR, variably scaled kernels, greedy or reduced basis algorithms, RBF-Partition of Unity and RBF-Finite Differences, to mention a few). Nevertheless, there are still intriguing interesting challenges to address: 1. theoretical advances in the above mentioned methods for PDEs approximation are needed; 2. the strong theoretical background already established for scattered data fitting can be leveraged to gain valuable insights in the context of scientific machine learning and explainable artificial intelligence; 3. explore new applied frontiers, such as kernel-based interpolation/extrapolation algorithms for inverse problems which are common to many (medical, astronomical, climate)-signal and image processing issues. The aim of this minisymposium is to bring together an international community of researchers with expertise in the theoretical, computational, and practical aspects of kernel methods in order to foster discussion on recent advances and stimulate future research in this rapidly evolving field.

Patrick Joly, Maryna Kachanovska, Zos Moitier

Numerical integration is one of the classic yet actively developing topics in numerical analysis. It plays an important role in many methods of numerical resolution of partial differential equations and integral equations. As mathematical models become more complex, new numerical integration tools are needed for their efficient resolution. A significant amount of modern research is focused on the efficient computation of integrals involving singular and highly oscillatory functions on manifolds, integrals over rough domains (e.g., domains with fractal boundaries), and/or with respect to non-standard measures, such as Hausdorff measure. The objective of this minisymposium is to present recent advances in this area.

Emanuele Arcese, Matthias Baray, Luca Desiderio

Many wave propagation phenomena can be modelled using Boundary Integral Equation (BIE) techniques, whose discretisations are known as Boundary Element Methods (BEM). Such methods only require the meshing of the boundary of the physical domain, making the simulation setup straightforward and lowering the number of degrees of freedom. The contribution opportunities in the BEM approaches are numerous due to the challenges that the method presents in some specific application scenarios. Currently, the most widely spread discretisation methods employ low-order basis functions, typically of order one or two, as well as low-order approximation of the geometry by means of planar elements. Nonetheless, analyses have indicated that high-order methods deliver better computational efficiency in many instances. The main goal of the Minisymposium is to bring together experts and young researchers in the field of numerical analysis, belonging to different international research groups, to discuss the most recent advances and current open challenges on high-order innovative BEM strategies for both time-harmonic and time-domain problems. Application areas covered include, but are not limited to, elastodynamics, acoustics and electromagnetism. The emphasis will be on high-order BEM, both recent and traditional ones: h-, p- and hp-BEM, curvilinear BEM, Isogeometric Analysis BEM (IgA-BEM), Plane Wave enriched Partition of Unity BEM (PW-PUBEM), virtual element approximation-based BEM, spectral BEM, etc.

Christian Klingenberg, Wasilij Barsukow

Multi-dimensional conservation laws possess many more phenomena than their one-dimensional counterparts, for example turbulent flow with vortices and non-trivial stationary states. At the same time, computational resources are limited, and refinement is particularly costly for multi-dimensional simulations. Many currently available numerical methods are only able to capture multi-dimensional phenomena only on very fine grids. This is because they add numerical diffusion that is rooted in one-dimensional thinking, and also because the fluxes are computed using Riemann problems. For subsonic flow, the latter even spoil the solution because of excessive viscosity and not they are not naturally asymptotic preserving for the Mach number going to zero. In this mini-symposium we shall discuss a new hybrid finite element finite volume method that achieves upwinding by locally evolving continuous data, instead of solving Riemann problems. Its degrees of freedom are cell averages, moments and point values located at cell interfaces, the latter being shared between adjacent cells. The solution has a globally continuous representation. The evolution of the averages is conservative, which means this is a finite volume method that is able to converge to the weak solution. This method was proposed by Phil Roe 10 years ago. He suggested a 3rd order method that can be found in the literature under the name of Active Flux. Since then it has been shown to achieve superior results on coarse grids, because it is preserves multi-dimensional structures of the flow. In recent developments of this method, the evolution of higher moments makes Active Flux arbitrarily high-order accurate. This mini-symposium gathers researchers that have contributed towards this new Active Flux method for multi-dimensional systems of conservation laws, in particular the full multi-dimensional Euler equations. We expect this mini-symposium to spur more activity in the exciting new development of an efficient new finite volume method for hyperbolic conservation laws.

Michael Dumbser, Firas Dhaouadi, Laura del Rio Martin, Ilya Peshkov

The modeling, analysis and numerical treatment of multiphase fluid dynamics provide several difficult problems treated in the past as well as in the very recent literature. Often, very problem-specific approaches revealing different mathematical structures are used, making it difficult to provide a unified treatment of these topics. This has motivated the continuous development of high-order numerical methods, that are able to tackle these challenges and allow to numerically solve the governing equations efficiently with high accuracy while also conserving their underlying mathematical structure. This minisymposium will focus on the latest developments and applications in this context, both at the theoretical and numerical level. The topics include but are not limited to 1) Modeling, analysis, and numerical simulation of multiphase flows 2) Design and numerical simulation of hyperbolic models of multiphase flows 3) Structure-preserving and thermodynamically compatible schemes for time-dependent PDEs 4) High-order numerical methods for multiphase flows 5) Efficient time-stepping methods

Alina Chertock, Alexander Kurganov

Nonlinear hyperbolic (systems of) PDEs arise in a wide variety of applications, both scientific and practical. Solving these systems numerically is a challenging task due to appearance of nonsmooth solutions that may include shock, contact, and rarefaction waves as well as other complicated structures resulting from the wave interaction. These solution features often cannot be efficiently and accurately captured by low-order numerical methods. This mini-symposium focuses on recent advances and development of high-order and adaptive numerical methods for nonlinear hyperbolic problems. High-order methods are essential for achieving accurate approximations on relatively coarse meshes one has to use in most of the practically interesting applications. High-order methods, however, often suffer from lack of efficiency as they rely on computationally expensive high-order nonlinear reconstructions and interpolations. Adaptive numerical techniques help to increase the efficiency of high-order methods by automatically detecting regions in the computational domain, where high accuracy is most needed---neighborhoods of shock and contact waves and other rough regions of the computed solutions. The presentations in this mini-symposium will both highlight the state-of-the-art in high-order and adaptive numerical methods and provide a platform for discussing open problems and future directions in the field.

Jerome Droniou, Lourenco Beirao da Veiga, Paola Antonietti, Daniele Di Pietro

Complicated models of partial differential equations require flexible and robust numerical methods to be accurately approximated. Flexibility can be found in the capacity of a scheme to accept meshes with generic polygonal/polyhedral elements, which allows for example for seamless local mesh refinement and easier capture of complex domain geometries. Flexibility can also relate to the various orders of accuracy a method can achieve, depending on the particular needs (dictated, e.g., by the application and/or the expected local behaviour of the solution): low order for a cheap coarse approximation, high order for a better description of steep behaviours. The robustness of the method can mean that the method produces approximations that remain accurate in extreme regimes of the model's parameters, and/or that it reproduces particular features of the continuous model -- from energy bounds to constraint preservation. The latter property is often linked to reproducing, at the discrete level, particular properties of the underlying differential operator -- such as their arrangement in the form of a Hilbert complex. This minisymposium will be dedicated to such questions. We will cover recent advances and perspectives on related subjects such as: polytopal methods (applicable on generic polygonal/polyhedral meshes), arbitrary-order methods, discrete complexes, robustness with respect to parameters, etc.

Stefano Bonetti, Francesca Bonizzoni, Mattia Corti, Franco Dassi, Ivan Fumagalli

The numerical solution of multi-physics problems is of crucial importance; indeed, phenomena with different spatial or temporal scales, the interaction of several physical laws, and objects with different materials and properties are frequently found in nature. The numerical discretization of these problems typically requires the following features: (i) accurate representation of complex geometries, (ii) ability to cope with intricate and/or non-conforming interfaces, (iii) robustness with respect to heterogeneities of physical properties. On top of that, flexible mesh generation and adaptation strategies, possibly combined with high-order approaches, can foster the efficiency and accuracy of the numerical method. Advanced numerical methods that cover most of these features are Polytopal Methods (such as Discontinuous Galerkin, Virtual Element, Hybrid High-Order, Hybridizable Discontinuous Galerkin, etc., based on a polygonal/polyhedral mesh), eXtended Finite Element Methods, p-, h-, and hp-adaptive FEM, multi-scale discretization schemes, CUTFEM, and others. The primary objective of this minisymposium is to explore recent advancements in state-of-the-art numerical methods and their application in the context of multi-physics problems. The minisymposium will focus (non-exclusively) on applications regarding biological tissues and processes, geological materials, fracture and contact mechanics, fluid-structure and fluid-porous-structure interaction, non-isothermal flows, and deformation processes.

Carlo Marcati, Christoph Schwab

Recent years have seen impact of deep learning approaches in scientific computing, science, and engineering. Methods based on deep learning have been used, for example, to approximate solutions and solution families of parametric PDEs. Particular attention has been paid to realizing approximation performance corresponding to spectral and high order methods, possibly in more general settings, with convergence rate guarantees. This minisymposium will showcase recent contributions to and progress on this theme, with particular attention on techniques in scientific machine learning for operator learning and operator representation.

Thophile Chaumont-Frelet, Markus Melenk

Numerical discretizations of wave propagation problems, be it in the high-frequency or long-time regime, typically suffer from dispersion errors that propagate throughout the computational grid. This makes the approximation of such wave propagation problems very challenging and computationally intensive. Besides, other aspects such as complex geometries, highly heterogeneous media, or nonlinear material lays might further complicate the task. This minisymposium is dedicated to the design, analysis and implementation of robust numerical methods for wave propagation problems. These encompass high-order finite element and discontinuous Galerkin methods, multiscale methods, boundary element methods and Trefftz methods.

Hojun You, Jin Seok Park, Jae-Hun Jung, Sehun Chun

High-order CFD methods offer distinct advantages in accuracy and computational efficiency compared to traditional lower-order techniques, making them particularly well-suited for capturing complex flow phenomena across various engineering applications. These methods are increasingly being adopted in diverse fields, including aerospace and mechanical engineering and bio-medical sciences, where resolving intricate multi-scale flow physics is crucial. Despite their promise, widespread adoption of high-order methods in industry faces several unresolved challenges. Key issues include developing stable methods for handling aliasing and shock-driven instabilities, flexible approaches for complex unstructured meshes, scalable and memory-efficient preconditioners, and optimizing computational kernels for modern high-performance computing platforms. This mini-symposium addresses recent advances in high-order CFD methods, emphasizing both numerical and computational strategies to enhance accuracy and efficiency in a wide range of engineering applications. Topics of interest include, but are not limited to, high-order spatial and temporal discretization, shock-capturing methods, moving and overset meshes, hp-adaptation, and high-performance computing for multi-scale engineering simulations. By bringing together researchers from both fundamental and applied CFD fields, this symposium will foster discussion, knowledge exchange, and an assessment of the latest innovations in high-order CFD methods, contributing to their continued evolution and practical application in challenging engineering domains.

David Del Rey Fernndez, Frank Gaitan, Ala Shayeghi

Quantum computing promises a new frontier in computational resources far beyond that achievable using classical computing hardware and, theoretically, can provide significant, up to exponential, speedup over classical computers in select tasks (e.g., the Harrow, Hassidim, and Lloyd). Only relatively recently has attention been shifted to investigate the development of quantum computing algorithms for linear and nonlinear partial differential equations (PDEs). However, obtaining quantum speedup is a delicate task requiring a careful analysis of all algorithmic components from state preparation, solution, to state tomography. Indeed, harnessing this power will require fundamentally new ways of approaching the discretization of PDEs (e.g., Carleman linearization, hybrid quantum-classical algorithms, etc.). Nevertheless, classical computing numerical analysis frameworks will provide the essential tools for developing these novel discretizations (e.g., stability, convergence, and error estimation) and in particular high-order frameworks are an attractive approach that can provide essential flexibility needed to squeeze performance out of quantum computing platforms. The objective of this mini symposium is to bring together numerical analysts working on a variety of approaches aimed at developing quantum algorithms for both linear and nonlinear PDEs.

David Del Rey Fernndez, Nathaniel Trask

Machine learning (ML) has found extraordinary success on tasks such as image recognition and natural language processing. In science and engineering, there is increased interest in applying ML broadly. For example, in the context of numerically solving partial differential equations, approaches such as physics informed neural networks have gained significant traction. More recently, there has been a push to incorporated ML directly in classical numerical methods frameworks and by doing so retain mathematical frameworks supporting stability, accuracy, and structure preservation analyses. However, much work remains to leverage the speed up benefits of ML while retaining the properties of classical algorithms, particularly in the context of nonlinear PDEs and complex geometry. In this mini-symposium the focus is broadly on ML techniques applicable to the numerical solution of nonlinear PDEs. We bring together a diverse set of researchers working on a variety of approaches aimed at having range of provable properties and that result in high-resolution schemes.

Anita Gjesteland, Jens Keim, Per-Olof Persson, Christian Rohde

Nonlinear hyperbolic balance laws are widely used for the modeling of fluid mechanical processes. However, fundamental questions about distinctive hyperbolic features such as the multi-scale interference of shock-waves and shear-instabilities or the interplay of hyperbolic transport and random environments remain vastly open and pose major challenges concerning their numerical treatment. One example is the widely-accepted entropy principle which has been found to be insufficient to point out unique solutions of classical continuum-scale systems but admits a multitude of complex violently-oscillating flow fields. Hence, this minisymposium is devoted to the discussion of recent developments and new avenues for the design of tailored numerical methods for hyperbolic balance laws. Therefore, contributions from the fields of novel solution concepts for hyperbolic models and corresponding numerical schemes, multi-scale models and methods as well as numerical approaches for probabilistic models in the field of fluid dynamics are welcome. This includes high-resolution numerical schemes based on, e.g., generalized entropy methods, dissipative limits or probabilistic as well as high-order methods for the solution of multi-scale or non-local problems such as numerical schemes to incorporate subgrid information into continuum scale simulations. Furthermore, recent advances from the field of high-order entropy preserving or stable schemes as well as structure preserving numerical methods which ensure well-balancing and the preservation of asymptotic states, e.g., in the presence of Mach number limits, are desired. Finally, numerical methods and uncertainty quantification approaches for stochastic models of hyperbolic systems arising in fluid mechanics are highly appreciated. This includes probabilistic modeling concepts to explore statistical turbulence using e.g. stochastic variational principles or the exploration of stochastic/data-driven tools for hybrid perturbation/filtering techniques. This minisymposium features on the one hand research pursued by national and internatial scientists associated to the German Priority Research Programme SPP-2410: Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness and on the other hand invites scientists from the community to contribute, exchange and discuss new avenues for the numerical treatment of hyperbolic balance laws.

Yongyong Cai, Lilian Wang

It is known the spectral method has high accuracy, while its performance depends on the structure of the underlying problem. For complex PDEs with variable coefficients, highly nonlinear terms and strong coupling, designing efficient spectral and high order methods is not an easy task. In addition, the performance of the spectral method might be deleteriously degraded when the solution exhibits local singular behaviors with very limited regularity. In practice, the singularity may occur in various scenarios such as PDEs in nonsmooth computational domains with sharp corners or with degenerate or discontinuous coefficients (complex mediums), non-matching boundary conditions, singular kernels or potentials, and non-differentiable nonlinear terms among others. \medskip On the other hand, high-order methods have proven to be the method of choice for PDEs with oscillatory solutions with wide applications in simulations of water waves in ocean dynamics, matter waves in quantum physics and soliton waves in optical fibers among others. and many of them can be described by PDEs such as nonlinear Scr\odinger equation, Klein-Gordon equation, Kortewegde Vries equation and Maxwell equation. Efficient and accurate numerical simulations for such PDEs are highly demanding in scientific and engineering computations. Spectral methods are capable of producing very accurate simulation results, and more importantly, they require a substantially smaller number of unknowns (even for engineering accuracy) when compared to their lower-order counterparts. Given these two classes of challenging problems, how to properly design spectral methods and conduct the related numerical analysis is a research topic of longstanding interest and worthy of deep investigation. This mini-symposium aims at bringing together numerical analysts and computational scientists to present their findings on recent advances in algorithm development and analysis of spectral methods for complex PDEs with inhomogeneous medium, oscillatory behaviours and singularities. The theme is to present some novel developments of spectral method and applications of spectral methods in solving these PDEs.

Philipp Trunschke, Martin Eigel

While high-dimensional functions are ubiquitous in science and engineering applications, their approximation remains a challenging task often plagued by the curse of dimensionality. To break this curse, methods must exploit the functions inherent structures, such as regularity, known compositional structure, and anisotropic or compositional sparsity. Due to these methods nonlinear nature, their training is challenging, and a dynamic systems view of these modern formats presents great potential in this context. The high dimensionality presents another challenge, and the quality of this approximation depends strongly on the sample used for estimating the occurring high-dimensional integrals. Obtaining provable error bounds and efficient algorithms is the subject of active research. This mini-symposium aims to gather experts to present recent advances in approximation and learning of high-dimensional functions.

Dmitri Kuzmin, Andrs Rueda-Ramrez

This minisymposium will bring together developers of structure-preserving and essentially non-oscillatory numerical methods for nonlinear hyperbolic (systems of) conservation laws and related problems. The main focus will be on achieving optimal accuracy in the context of stabilized high-order methods that guarantee entropy stability, preservation of invariant domains, validity of local discrete maximum principles and/or lack of spurious oscillations in the quantities of interest. The stabilization approaches to be presented and discussed include (subcell versions of) convex limiting techniques, nonlinear artificial viscosity methods, accuracy-preserving smoothness sensors, entropy fixes, and combinations thereof. We also welcome presentations on appropriate formulations of inequality constraints, limiters for high-order time discretizations, computation of steady-state solutions, and bound-preserving schemes that use local time stepping or adaptive mesh refinement.

Tamas Horvath, Tan Bui-Thanh

Complex multiphysics systems appear in various critical real-life applications, including multiphase flows, computational geosciences, magnetohydrodynamics, fluid-structure interactions, etc. These highly nonlinear problems describe strongly coupled physical mechanisms that interact across a wide range of length and time scales, necessitating robust and efficient high-resolution numerical approximations. Therefore, developing accurate and effective methods that leverage parallel computation at extreme scales is crucial. Numerical discretizations and solvers for practical multiphysics simulations must meet the following criteria: High-order accuracy in space and time; Stability; Conservativeness; Minimal degrees of freedom for implicit solution approaches; (5) Suitability for unstructured meshes; (6) Compatibility with hp-adaptivity; (7) Effectiveness in handling disparate temporal and spatial scales; and (8) Optimization for fine-grain parallelism. This minisymposium provides a platform for researchers to present novel high(er)-order numerical methods for solving nonlinear multiphysics systems. The talks will address theoretical, numerical, and computational issues critical to developing approaches that exhibit these desired properties.

Oscar Bruno, Mark Lyon

Despite their ubiquity in diverse applications (radar, sonar, communications, imaging, etc.) and developments in algorithms and computation power, many wave propagation problems remain intractable while the solutions to others are not efficient enough for practical application. Presentations will focus on a variety of new solution methodologies and other advances that have been recently introduced to significantly increase our capability to solve the most difficult of these important problems, including the development of new high order solvers and algorithms, novel methods for the treatment of problems of electromagnetic scattering by domains with corners, methodologies designed to solve for time-dependent wave propagation on the basis of frequency domain solutions, methods for computing resonances and spectra, methods to account for inhomogeneity and layered media, solver acceleration techniques, and developments in optimization including both accuracy and efficiency optimization.

Jose E Castillo, Miguel Dumett, Anand Srinivasan, Jarred Brzenski

High Order Mimetic Differences methods are based on Mimetic Operators which are discrete analogs of the first-order invariant operators Divergence, Gradient, and Curl. These Operators have been shown to satisfy all the vector calculus identities in the discrete sense. In addition, they have the same order of approximation in the interior of the domain as well as the boundary. In this section, we will discuss shock-capturing capabilities, linear and nonlinear conservation laws, skew-symmetric formulation to handle the nonlinear convective term in the incompressible Navier Stokes equations as well as the computational efficiency of solving these equations with High Order Mimetic Differences.

Kenneth Duru, Kieran Ricardo, David Lee, Tom Hagstrom

Robust and high order accurate numerical methods for numerical simulations of PDEs have increasingly become an appealing choice for several modelling applications, owing to their efficiency and scalable performance on modern supercomputers. However, the design of robust and reliable high order accurate numerical methods for nonlinear PDEs pose a significant challenge, as initial attempts often result in crashes due to compounding numerical errors or the presence of undesirable numerical oscillations which can pollute numerical simulations everywhere. Desirable high order accurate methods for nonlinear PDEs must be robust (provably stable) and preserve several important invariant present in the system. Ideally numerical methods should mimic the properties of the continuous system they approximate, but continuous {nonlinear } systems have infinitely many invariant of which only a finite number can be preserved by any discretization. To minimize the effects of numerical artifacts from contaminating results of numerical simulations it is desirable that numerical methods preserve some important invariant present in the physical model. For a target modelling application this necessitates choosing a subset of the invariant, based on the particular modelling application, for the discrete model to preserve. While discrete structure preservation of conservation laws and model symmetries of nonlinear PDEs can improve accuracy, mitigate against numerical instabilities and enhance the robustness of high order discretizations, this also introduces additional theoretical and computational demands, such as proof of nonlinear stability, mimetic reformulations and discretizations, well conditioned implicit time steppers and complex numerical fluxes. This mini-symposium will cover presentations on recent advances on robust and structure preserving high order numerical methods for PDEs. These will include, but not limited to, summation by parts and discontinuous Galerkin methods, and mixed and compatible finite element methods.

Michael Hecht, Phil-Alexander Hofmann, Gentian Zavalani

This minisymposium focuses on presenting efficient and fast numerical methods for computing function expansions with high approximation power, addressing key challenges in modeling complex, high-dimensional systems. The methods discussed are designed to enhance the performance of computational tasks such as solving PDEs, ODEs, and signal processing problems on both flat and manifold domains. Specifically, we will explore high-order (pseudo) spectral, kernel-based, and integration techniques that overcome the traditional computational bottlenecks arising in these contexts. While the Fast Fourier Transform (FFT) undeniably belongs to the greatest achievements in computational sciences of the last century, solving part of these challenges, this minisymposium discusses extensions and alternative methods that surpass the limitations of the FFT, including the requirement for periodicity, specifically (uniformly) structured collocation points, and its sensitivity to the curse of dimensionality.

Pasquale Claudio Africa, Federico Pichi, Niccol Tonicello, Michele Girfoglio, Gianluigi Rozza

Over the past few decades, the use of numerical solvers in simulating complex physical phenomena has grown rapidly. Such solvers are especially valuable in fields like aerospace engineering, where accurate modeling of airflows is critical to optimizing aircraft performance, and in the biomedical sector, where blood flow simulations are vital for developing medical devices. In particular, the use of high-order methods, such as spectral methods and Discontinuous Galerkin (DG) methods, offer superior accuracy compared to traditional low-order methods by using higher-degree polynomial approximations within each element of the computational mesh. These methods are rapidly spreading across various fields of Computational Fluid Dynamics (CFD), ranging from compressible fluid dynamics to combustion, multi-phase flows, and fluid-structure interaction. However, while high-order methods can often achieve accuracy and reliability, classical solvers' computational cost can be extremely demanding, limiting real-time and many-query simulations, which are of great interest for industrial and biomedical applications [7]. The use of advanced surrogate modeling techniques can significantly reduce the computational cost of CFD simulations without compromising the underlying accuracy. More specifically, Reduced Order Methods (ROMs) aim to build reliable and efficient models to study complex parametric systems, featuring an exponential approximation rate under suitable assumptions. Significant results have been achieved for both intrusive and non-intrusive ROMs, which balance physics-based and data-driven approaches. Recently, hybrid and machine-learning techniques have shown promising results, enhancing high-fidelity solvers and accelerating computations for Reynolds-Averaged Navier-Stokes and Large-Eddy Simulations models, multi-fidelity approaches, data-driven discretizations and reconstructions, and h/p automatic refinement. This minisymposium aims to bring together researchers from academia and industry who are active in developing and implementing surrogate modeling techniques to accelerate high-order numerical solvers, fostering collaboration and innovation in this rapidly evolving field.

Huiyuan Li, Zhiguo Yang

High-fidelity numerical methods are preferable for simulating large scale and complex physical systems, owing to their remarkable efficiency and accuracy in capturing intricate phenomena. This mini-symposium focuses on novel mathematical modeling, numerical approximation, parallel algorithm, efficient solvers in fluid dynamics, electromagnetism, kinetics, material science, ocean and marine science, neutron transport, etc.; It aims to bring together active researchers to present and discuss their recent advancements in high-fidelity methods such as spectral methods and structure-preserving methods. We will also explore their numerical analyses, software and applications together with their performance optimization across a variety of (multi-) physical problems in various disciplines and engineering fields.

Ludvig af Klinteberg, Fredrik Fryklund

The field of integral equation-based methods has advanced considerably in recent years. With new developments in theory, software, and algorithms, it is now mature enough for the community to tackle emerging challenges. These challenges include, but are not limited to, accurate quadrature in three spatial dimensions, complicated time-dependent PDEs, and identifying new real-world applications. The goal of this minisymposium is to bring together early career scientists to discuss cutting-edge research, with a focus on fast algorithms for tackling these issues. We will discuss future research directions and promote interdisciplinary research collaboration between computational scientists and other fields.

Anita Gjesteland, Zelalem Worku, Jesse Chan

Non-linear partial differential equations (PDEs) and their numerical approximations remain a popular field of research due to their importance in modelling real-life phenomena. The non-linear structure of the equations poses inherent challenge in developing provably stable schemes and continues to warrant further research. In recent years, discretizations that preserve different properties of the continuous equations, such as entropy stability and positivity (of certain variables), have shown promising results. Non-linear stability estimates can yield certain bounds on the solution which are useful in the study of well-posedness. Extension of such formulations for various non-linear PDEs, mathematical analysis at the discrete level, and numerical simulations of practically relevant problems remain important research avenues. The focus of this minisymposium is on recent advances in high-order provably entropy-stable discretizations of various non-linear PDEs and their applications to illuminate numerical challenges and potential areas for improvement. Relevant topics include, but are not limited to, analysis of entropy-stable spatial discretizations, positivity preservation, error estimation, shock capturing, time integration, efficient implementations, and numerical simulations.

Walter Boscheri, Francesco Fambri, Maria Han Veiga, Raphal Loubre, Vincent Perrier

Some hyperbolic systems are known to include implicit differential constraints, also called involutions. This can be for example the conservation of the vorticity for the wave system (linked with the low Mach number accuracy problem), the conservation of the divergence of the magnetic field for the Maxwell system or the Magnetohydrodynamics system, the curl of the deformation tensor for the hyperelastic system. These involutions are additional constraints with respect to the conservation laws. Numerical schemes that rely on the direct discretization of the conservation laws usually fail to ensure these involutions, and several strategies were historically proposed: Projection methods consist in a predictor step, in which the conservation law are advanced in time, and a corrector step in which the unknownw are projected so as to ensure the involutions. Generalized Lagrangian Multiplier methods consist in considering the constraints to preserve as additional variables, and in augmenting the system with additional equations which include relaxation terms that ensure asymptotic consistency with not only the initial system but also the involution. Staggered schemes rely on a smart positioning of the unknowns on the different entities of the mesh (vertices, edges, faces or cells) for automatically ensuring the preservation of involutions. The aim of this Minisymposium is to review a large spectrum of recent advances in high order methods for the conservation of involutions for hyperbolic conservation laws and to present examples and applications e.g. to electromagnetics, fluid mechanics and structural mechanics.

Charles William Parker, Pablo Brubeck

Many physical systems conserve some kind of quantity, such as mass and helicity in fluid flow to charge and current in electromagnetism. Structure-preserving discretizations aim to conserve some or all of these quantities and also often lead to stable methods. However, constructing structure-preserving discretizations is a nontrivial task, and high-order methods are often needed to avoid spurious behaviour or unnatural restrictions on the method, such as the need to use meshes with specific structure. In this minisymposium, we will focus on recent advancements of structure-preserving finite element methods for a variety of applicactions. Presentations will include the development of new methods, the analysis of new and existing methods, implementation aspects such as efficient algorithms and preconditioning, and novel applications.

Carlos A. Pereira, Shoyon Panday, Stphane Gaudreault, Philipp Birken

Recent advances in high-order methods have shown potential for application in numerical weather prediction (NWP) due to their inherent low-dissipative characteristic and its suitability to implementations in modern computing architectures. They are likely to serve as the base for next-generation weather forecasting systems. In addition, a wealth of research has focused on high-order time integration schemes, with many innovative approaches developed in recent years that enhance both accuracy and overall stability. In this mini-symposium, we aim to highlight discussions on the application of efficient, high-order spatial and time-integration schemes in numerical weather prediction using classical numerical methods, including current challenges, and potential solutions to address them.

Misun Min, Paul Fischer, Tzanio Kolev

High-order discretizations have the potential to provide an optimal strategy for achieving high performance and delivering fast, efficient, and accurate simulations on next-generation architectures. This minisymposium will discuss the next-generation high-order discretization algorithms, based on finite/spectral element approaches that will enable a wide range of important scientific applications to run efficiently on future architecture. Topics will include high-order discretization kernels and lightweight portable libraries that are critical for performance and exascale applications as well as algorithm and software of arbitrarily high order to impact the design of exascale architectures, system and application software for improved portability and performance of the high-order methods.

Manuel Sanchez, Jeonghun Lee

This mini-symposium focuses on recent advancements in structure-preserving high-order numerical methods and techniques. These methods are designed to maintain the underlying structure of physical systems. The contributions encompass a wide range of topics, including symplectic and energy-preserving time integration methods, Hamiltonian structure-preserving methods, geometric approaches, advanced finite element methods, space-time methods, and wave propagation techniques. The primary focus is on how these methods retain essential physical properties of the original continuous systems, thereby enhancing accuracy, stability, and computational efficiency. The symposium welcomes contributions that feature new theoretical developments, numerical algorithms, and applications in science and engineering.

James McDonald, Clinton Groth

Many multiscale modelling situations are well described by kinetic equations. These fields include rarefied gas flow, plasma flows, polydisperse multiphase flows, radiative transfer, quantum mechanics, and many more. The relevant similarity between these phenomena is that they involve a very large number of discrete particles or elements. The sheer number of particles precludes a direct numerical treatment. Therefore, phase-space density methods are often adopted in the form of kinetic models. Typically, kinetic models for these cases are very high dimensional. This high dimensionality imposes prohibitive computational cost on naive numerical solution techniques in these cases. The high dimensionality of these models can often be alleviated through the use of techniques, such as moment closures, that trade the high dimensionality for an expanded solution vector describing the evolution of statistics of the collection. These models typically take the form of non-linear differential or intergo-differential balance laws with stiff local source terms. The accurate and efficient numerical resolution of such models requires specially designed spectral or high-order techniques. The aim of this minisymposium is to assemble researchers in the area of high-order/spectral numerical simulation of kinetic or moment equations in order to discuss recent developments, to explore open issues, and to foster cross-fertilization. Specifically, high-order or spectral methods for the direct numerical solution of the possibly high-dimensional kinetic equations and modern high-order or spectral methods for the accurate resolution of moment-closure models are of interest.

Nour Al Hassanieh, Tristan Goodwill

The study and simulation of wave propagation are essential to understanding many physical phenomena and engineering problems. Due to the highly oscillatory behaviors, wave propagation problems can be challenging to solve accurately and efficiently. This is especially true in intricate geometries, where repeated reflections can slow down the convergence of algorithms. High order and spectral methods are particularly important for wave problems as they use fewer unknowns per oscillation while maintaining high accuracy. In this mini-symposium, we discuss several new fast, high-order and spectral methods for solving oscillatory PDEs in complex domains. Examples of problems will include the wave equation, Schrdingers equation, the time harmonic Maxwells equations, and more. The discussed methods will include integral equation formulations, the method of fundamental solutions, spectral domain decomposition, and Fourier methods.

Catherine Mavriplis, Paul Fischer, Masayuki Yano

Anthony Pateras research career has profoundly shaped the landscape of high-order numerical methods, including several pivotal advancements. His early work with Steven Orszag on spectral methods contributed precision and new insights to the theory of hydrodynamic stability of canonical flows, paving the way for further investigations into more practical flow phenomena. Pateras invention of the spectral element method marked a significant evolution, providing a viable framework for high-accuracy simulations of complex engineering flows and extending its impact into diverse domains such as solid mechanics, wave propagation, magnetohydrodynamics and, ocean and atmospheric modeling. His early idea to develop a multipurpose code, first known as Nekton and later developed as Nek5000, and today NekRS on GPUs, has catapulted the high order communitys credibility as codes are now competitive with or better than more traditional methods and regularly produce Direct Numerical Simulations of extremely complex engineering flows in high performance computing environments, including a route to exascale computing. Turning his awesome mathematical skills to reduced basis methods, he developed important mathematical underpinnings for the field, including robust error estimation, ensuring that the models remain reliable. In both fields, he used his keen mathematical sense to develop efficient algorithms that made the methods viable and applicable to efficient parameter estimation, design, optimization, and real-time control (for the case or reduced basis modeling). His many close partnerships with mathematicians and engineers have been especially fruitful, providing a key bridge between theory and practical implementation. This symposium will feature 12-16 talks on entire fields that have developed out of the pioneering work of Anthony Patera in both, high order methods for efficient high accuracy engineering calculations, and reduced basis methods for real time computing.

Jiaxi Gu, Jae-Hun Jung

The neural network approach has been actively studied in the context of classical PDE solvers and has proven its potential and efficiency. This minisymposium focuses on recent developments in neural network techniques, particularly for the WENO and dG methods. For these methods, the minisymposium highlights neural network research addressing key problems such as discontinuity detection, efficient determination of nonlinear weights, reduction of computational complexity, and boundary preservation, etc. In this minisymposium, we aim to bring together researchers who have developed new neural network methods for these problems in both WENO and dG methods.

Qing Cheng, Jie Shen, Jiang Yang, Fukeng Huang

Complex systems which appear to be nonlinear partial differential equations (PDEs) have been widely used in various fields, such as thermodynamics, biology, material science, electromagnetism, to name just a few. Even though the history of the study on numerical PDEs is quite long, there are still many open and important questions. In this mini-symposium, we aim at gathering researchers working on the topic to discuss recent advances on the development of structure-preserving numerical methods and numerical analysis of high-order numerical methods for approximately solving nonlinear complex systems, in order to further promote the developments of the topic.

Zhaopeng Hao, Xiaobo Yin

Nonlocal models (including fractional PDEs, nonlinear partial integrodifferential equations in kinetic theory, and general nonlocal PDEs that arise in Peridynamics) that account for interaction occurring at a distance have been shown to provide improved simulation fidelity in the presence of long-range forces and anomalous behaviors. However, these models numerical discretization faces more challenges than the traditional local model. This mini-symposium provides a platform for researchers to share their experiences and insights on designing robust and efficient numerical schemes for nonlocal models.

William Alvah Sands, Jing-Mei Qiu

The numerical solution of high-dimensional partial differential equations (PDEs) presents substantial challenges for traditional solvers due to the severe storage demands imposed by the curse of dimensionality (CoD). This bottleneck is particularly significant in fields such as quantum mechanics, astrophysics, fusion, and fluid dynamics, where high-dimensional PDEs are frequently encountered. Low-rank methods have recently gained traction as a promising approach to mitigate the CoD by representing solutions through efficient factorizations that expose low-rank structures, leading to substantially smaller storage and computational requirements. This minisymposium aims to bring together researchers developing innovative low-rank methods based on high-order discretizations to discuss algorithms that can substantially improve the efficiency and accuracy of solving high-dimensional problems in a variety of applications.

Carolyn M. V. Pethrick, Mohammad R. Najafian

Recent research efforts have developed high order spatial discretizations that are increasingly capable of efficiently predicting high-fidelity, unsteady flows. High order methods are well-suited to the complicated and unsteady physics found in many problems of industrial interest. Despite developments improving the scalability and robustness of high order semidiscretizations, efficient temporal integration remains an open problem. Presently, many approaches have been proposed to develop a temporal integration scheme that upholds the stability properties of the spatial method, maintains parallelizability, and minimizes computational cost. This minisymposium will bring together research efforts surrounding novel temporal integration strategies. Topics include but are not limited to: implicit and IMEX temporal integration; stability considerations, especially entropy stability and SSP; space-time approaches; spatially-partitioned temporal integration strategies, including P-ERK schemes and multirate time-stepping; adaptive time stepping; and applications of novel temporal integration methods to large-scale problems.

Brian Vermeire, Freddie Witherden, Peter Vincent

After nearly 20 years, flux reconstruction methods have become a wide-spread framework for recovering existing high-order schemes and creating a continuous range of others. This mini symposium focuses on the development and application of flux reconstruction methods. Theoretical developments include, but are not limited to, novel time stepping methods, linear and non-linear stability, extended ranges of flux reconstruction schemes, shock capturing, adaptivity, implicit large eddy simulation, new systems of governing equations, amongst others. Industrial applications include, but are not limited to, the development of new implementations of the flux reconstruction method and their application to industrial application to a wide range of applications. It is expected that this mini symposia will bring together leading experts on the development and use of flux reconstruction methods.

Jason Kaye

Tremendous progress in the control of quantum materials and devices has driven a need for high-precision computational methods capable of reliably reproducing and predicting experimental findings. This is a relatively open frontier in applied and computational mathematics, covering both effective non-interacting methods such as density functional theory and, increasingly, many-body methods applicable to systems with strong electron interactions. High-order methods, traditionally developed for the differential and integral equations of classical physics, have become a valuable component in robust, accurate, and high-performance algorithms for computational quantum physics. This minisymposium will showcase recent advances in this area, with applications including electron structure, many-body methods in and out of equilibrium, and open quantum systems.

Jason Hicken, Graeme Kennedy

High-order methods are generally more efficient than low-order methods. The efficiency of high-order methods makes them attractive for meta-analyses, such as partial-differential-equation (PDE) constrained optimization. A typical PDE-constrained optimization requires tens, if not hundreds, of simulations over a range of parameter values, so the faster runtime of high-order methods (for the same accuracy) can translate into significant time savings. Furthermore, compared to low-order methods, high-order methods are less susceptible to the optimization exploiting discretization errors to artificially improve the objective. The goal of this minisymposium is to showcase how high-order methods can be used in the context of PDE-constrained optimization. It will also highlight some of the challenges that arise in the implementation of high-order methods for optimization, and how those challenges can be addressed.