Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well

Aggregation-diffusion equations: dynamics, asymptotics, and singular limits

Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past fifteen years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blowup. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, as well as localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method -- the blob method for diffusion -- to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Nonlinear Hyperbolic Problems: modeling, analysis, and numerics

The workshop gathered together leading international experts, as well as most promising young researchers, working on the modelling, the mathematical analysis, and the numerical methods for nonlinear hyperbolic partial differential equations (PDEs). The meeting focussed on addressing outstanding issues and identifying promising new directions in all three fields, i.e. modelling, analysis, and numerical discretization. Key questions settled around the lack of well-posedness theories for multidimensional systems of conservation laws and the use of hyperbolic modelling beyond the classical topic of gas dynamics. A focal point in numerics has been the discretization of random evolutions and uncertainty quantification. Equally important, new multi-scale methods and schemes for asymptotic regimes have been considered

Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

Modelling and numerical analysis of energy-dissipating systems with nonlocal free energy

The broad objective of this thesis is to design finite-volume schemes for a family of energy-dissipating systems. All the systems studied in this thesis share a common property: they are driven by an energy that decreases as the system evolves. Such decrease is produced by a dissipation mechanism, which ensures that the system eventually reaches a steady state where the energy is minimised. The numerical schemes presented here are designed to discretely preserve the dissipation of the energy, leading to more accurate and cost-effective simulations. Most of the material in this thesis is based on the publications [16, 54, 65, 66, 243]. The research content is structured in three parts. First, Part II presents well-balanced first-, second- and high-order finite-volume schemes for a general class of hydrodynamic systems with linear and nonlinear damping. These well-balanced schemes preserve stationary states at machine precision, while discretely preserving the dissipation of the discrete free energy for first- and second-order accuracy. Second, Part III focuses on finite-volume schemes for the Cahn-Hilliard equation that unconditionally and discretely satisfy the boundedness of the phase eld and the free-energy dissipation. In addition, our Cahn-Hilliard scheme is employed as an image inpainting filter before passing damaged images into a classification neural network, leading to a significant improvement of damaged-image prediction. Third, Part IV introduces nite-volume schemes to solve stochastic gradient-flow equations. Such equations are of crucial importance within the framework of fluctuating hydrodynamics and dynamic density functional theory. The main advantages of these schemes are the preservation of non-negative densities in the presence of noise and the accurate reproduction of the statistical properties of the physical systems. All these fi nite-volume schemes are complemented with prototypical examples from relevant applications, which highlight the bene fit of our algorithms to elucidate some of the unknown analytical results.Open Acces