Author: Publisher: ISBN: Category : Languages : en Pages : 318
Book Description
Hyperbolic and kinetic equations often have parameters that vary considerably over the region. In certain asymptotic regimes where the parameter is very small, the standard hyperbolic or kinetic solvers break down because of the prohibitive computational cost. This thesis explores two efficient methods --- Domain Decomposition methods and Asymptotic Preserving (AP) methods for these problems. The first part aims at constructing a domain decomposition formulation for the Jin-Xin relaxation system with two-scale relaxations, which is a prototype for more general physical problems such as phase transitions, river flows, kinetic theories etc.. We propose the interface condition based on the sign of the characteristic speed at the interface. A rigorous analysis on the L2 error estimate is presented, based on the Laplace Tranform, for the linear case with an optimal convergence rate. For the nonlinear case, using standard compactness argument, we are able to prove the asymptotic convergence of the solution of the original relaxation system to the unique entropy weak solution of the domain decomposition system. The interface condition is derived rigorously by matched asymptotic analysis for a general flux with an extension to the case when a standing shock is sticking to the interface. The second part focuses on the development of AP methods for kinetic equations in the high field regime where both the collision and field effect dominate the evolution. The stiff force term poses extra numerical challenges as apposed to the stiff collision term which has been well-studied in the hydrodynamic regime. We first consider the Vlasov-Poisson-Fokker-Planck system used in electrostatic plasma and astrophysics. The AP scheme is constructed based on the combination of two stiff terms so as to use the symmetric discretization. The semiconductor Boltzmann equation is considered next. By penalizing the collision term by a classical BGK operator and treating the force term implicitly, we are able to overcome the exceptional difficulty that no specific expression of the local equilibrium is available. The distribution function is still shown to converge to the high field limit, which guarantees the capturing of the asymptotics without numerically resolving the small parameter.
Author: Giacomo Albi Publisher: Springer Nature ISBN: 3031298756 Category : Mathematics Languages : en Pages : 241
Book Description
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.
Author: Stéphane Cordier Publisher: European Mathematical Society ISBN: 9783037190128 Category : Differential equations, Hyperbolic Languages : en Pages : 372
Book Description
Hyperbolic and kinetic equations arise in a large variety of industrial problems. For this reason, the Summer Mathematical Research Center on Scientific Computing and its Applications (CEMRACS), held at the Center of International Research in Mathematics (CIRM) in Luminy, was devoted to this topic. During a six-week period, junior and senior researchers worked full time on several projects proposed by industry and academia. Most of this work was completed later on, and the present book reflects these results. The articles address modelling issues as well as the development and comparisons of numerical methods in different situations. The applications include multi-phase flows, plasma physics, quantum particle dynamics, radiative transfer, sprays, and aeroacoustics. The text is aimed at researchers and engineers interested in applications arising from modelling and numerical simulation of hyperbolic and kinetic problems.
Author: Shi Jin Publisher: Springer ISBN: 3319671103 Category : Mathematics Languages : en Pages : 282
Book Description
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.
Author: Remi Abgrall Publisher: Elsevier ISBN: 044463911X Category : Mathematics Languages : en Pages : 612
Book Description
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
Author: Elena Vázquez-Cendón Publisher: CRC Press ISBN: 020356233X Category : Mathematics Languages : en Pages : 434
Book Description
Numerical Methods for Hyperbolic Equations is a collection of 49 articles presented at the International Conference on Numerical Methods for Hyperbolic Equations: Theory and Applications (Santiago de Compostela, Spain, 4-8 July 2011). The conference was organized to honour Professor Eleuterio Toro in the month of his 65th birthday. The topics cover
Author: Remi Abgrall Publisher: Elsevier ISBN: 0444637958 Category : Mathematics Languages : en Pages : 668
Book Description
Handbook of Numerical Methods for Hyperbolic Problems explores the changes that have taken place in the past few decades regarding literature in the design, analysis and application of various numerical algorithms for solving hyperbolic equations. 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 readily understand their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Ideal for readers working on the theoretical aspects of algorithm development and its numerical analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or readers involved in applications Written by leading subject experts in each field who provide breadth and depth of content coverage
Author: Elena Vázquez-Cendón Publisher: CRC Press ISBN: 0203590627 Category : Mathematics Languages : en Pages : 144
Book Description
This volume contains the lecture notes of the Short Course on Numerical Methods for Hyperbolic Equations (Faculty of Mathematics, University of Santiago de Compostela, Spain, 2-4 July 2011). The course was organized in recognition of Prof. Eleuterio Toro‘s contribution to education and training on numerical methods for partial differential equation
Author: María Luz Muñoz-Ruiz Publisher: Springer Nature ISBN: 3030728501 Category : Mathematics Languages : en Pages : 269
Book Description
The present volume contains selected papers issued from the sixth edition of the International Conference "Numerical methods for hyperbolic problems" that took place in 2019 in Málaga (Spain). NumHyp conferences, which began in 2009, focus on recent developments and new directions in the field of numerical methods for hyperbolic partial differential equations (PDEs) and their applications. The 11 chapters of the book cover several state-of-the-art numerical techniques and applications, including the design of numerical methods with good properties (well-balanced, asymptotic-preserving, high-order accurate, domain invariant preserving, uncertainty quantification, etc.), applications to models issued from different fields (Euler equations of gas dynamics, Navier-Stokes equations, multilayer shallow-water systems, ideal magnetohydrodynamics or fluid models to simulate multiphase flow, sediment transport, turbulent deflagrations, etc.), and the development of new nonlinear dispersive shallow-water models. The volume is addressed to PhD students and researchers in Applied Mathematics, Fluid Mechanics, or Engineering whose investigation focuses on or uses numerical methods for hyperbolic systems. It may also be a useful tool for practitioners who look for state-of-the-art methods for flow simulation.
Author: 
Author: Giacomo Albi
Author: Stéphane Cordier
Author: Shi Jin
Author: Remi Abgrall
Author: Elena Vázquez-Cendón
Author: Remi Abgrall
Author: Elena Vázquez-Cendón
Author: María Luz Muñoz-Ruiz
Author: Alexandre L. Madureira