Uncertainty Quantification For Hyperbolic And Kinetic Equations

This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.

A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021.

The book originates from the mini-symposium "Mathematical descriptions of traffic flow: micro, macro and kinetic models" organised by the editors within the ICIAM 2019 Congress held in Valencia, Spain, in July 2019. The book is composed of five chapters, which address new research lines in the mathematical modelling of vehicular traffic, at the cutting edge of contemporary research, including traffic automation by means of autonomous vehicles. The contributions span the three most representative scales of mathematical modelling: the microscopic scale of particles, the mesoscopic scale of statistical kinetic description and the macroscopic scale of partial differential equations.The work is addressed to researchers in the field.

This monograph presents computational techniques and numerical analysis to study conservation laws under uncertainty using the stochastic Galerkin formulation. With the continual growth of computer power, these methods are becoming increasingly popular as an alternative to more classical sampling-based techniques. The text takes advantage of stochastic Galerkin projections applied to the original conservation laws to produce a large system of modified partial differential equations, the solutions to which directly provide a full statistical characterization of the effect of uncertainties. Polynomial Chaos Methods of Hyperbolic Partial Differential Equations focuses on the analysis of stochastic Galerkin systems obtained for linear and non-linear convection-diffusion equations and for a systems of conservation laws; a detailed well-posedness and accuracy analysis is presented to enable the design of robust and stable numerical methods. The exposition is restricted to one spatial dimension and one uncertain parameter as its extension is conceptually straightforward. The numerical methods designed guarantee that the solutions to the uncertainty quantification systems will converge as the mesh size goes to zero. Examples from computational fluid dynamics are presented together with numerical methods suitable for the problem at hand: stable high-order finite-difference methods based on summation-by-parts operators for smooth problems, and robust shock-capturing methods for highly nonlinear problems. Academics and graduate students interested in computational fluid dynamics and uncertainty quantification will find this book of interest. Readers are expected to be familiar with the fundamentals of numerical analysis. Some background in stochastic methods is useful but notnecessary.

In recent decades, kinetic theory - originally developed as a field of mathematical physics - has emerged as one of the most prominent fields of modern mathematics. In recent years, there has been an explosion of applications of kinetic theory to other areas of research, such as biology and social sciences. This book collects lecture notes and recent advances in the field of kinetic theory of lecturers and speakers of the School “Trails in Kinetic Theory: Foundational Aspects and Numerical Methods”, hosted at Hausdorff Institute for Mathematics (HIM) of Bonn, Germany, 2019, during the Junior Trimester Program “Kinetic Theory”. Focusing on fundamental questions in both theoretical and numerical aspects, it also presents a broad view of related problems in socioeconomic sciences, pedestrian dynamics and traffic flow management.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage

This contributed volume investigates several mathematical techniques for the modeling and simulation of viral pandemics, with a special focus on COVID-19. Modeling a pandemic requires an interdisciplinary approach with other fields such as epidemiology, virology, immunology, and biology in general. Spatial dynamics and interactions are also important features to be considered, and a multiscale framework is needed at the level of individuals and the level of virus particles and the immune system. Chapters in this volume address these items, as well as offer perspectives for the future.