Modeling in Applied Sciences

by Bellomo, Nicola; Pulvirenti, Mario

• ISBN13: 9780817641023
• ISBN10: 0817641025
• Format: Hardcover
• Copyright: 2000-05-01
• Publisher: Birkhauser

Summary

Modeling complex biological, chemical, and physical systems, in the context of spatially heterogeneous mediums, is a challenging task for scientists and engineers using traditional methods of analysis. Modeling in Applied Sciences is a comprehensive survey of modeling large systems using kinetic equations, and in particular the Boltzmann equation and its generalizations. An interdisciplinary group of leading authorities carefully develop the foundations of kinetic models and discuss the connections and interactions between model theories, qualitative and computational analysis and real-world applications. This book provides a thoroughly accessible and lucid overview of the different aspects, models, computations, and methodology for the kinetic-theory modeling process. Topics and Features: * Integrated modeling perspective utilized in all chapters * Fluid dynamics of reacting gases * Self-contained introduction to kinetic models * Beckera??Doring equations * Nonlinear kinetic models with chemical reactions * Kinetic traffic-flow models * Models of granular media * Large communication networks * Thorough discussion of numerical simulations of Boltzmann equation This new book is an essential resource for all scientists and engineers who use large-scale computations for studying the dynamics of complex systems of fluids and particles. Professionals, researchers, and postgraduates will find the book a modern and authoritative guide to the topic.

Table of Contents

Preface	xiii
Generalized Kinetic Models in Applied Sciences	1	(20)
N. Bellomo
M. Pulvirenti
Introduction	1	(2)
The Boltzmann Equation	3	(7)
The Vlasov or Mean-Field Equation	10	(1)
Generalized Kinetic Models	11	(2)
Generalized Models and Plan of the Book	13	(5)
References	18	(3)
Rapid Granular Flows: Kinetics and Hydrodynamics	21	(60)
I. Goldhirsch
Introduction	21	(3)
One Dimensional Hydrodynamics	24	(7)
Introductory remarks	24	(1)
The system	25	(1)
Homogeneous dynamics: Mean field results	25	(2)
Hydrodynamic equations	27	(4)
The Two Dimensional Case: Stationary Shear Flow	31	(11)
Introduction	31	(1)
Formulation of the problem	32	(1)
Perturbative expansion	33	(2)
The first order term	35	(2)
The second order term	37	(2)
The stress tensor	39	(2)
Summary of Section 2.3	41	(1)
The Unsteady Two Dimensional Case	42	(6)
Introduction	42	(1)
Formulation of the problem	42	(6)
The Three Dimensional Case: Hydrodynamic Equations	48	(13)
Introduction	48	(1)
Formulation of the problem	49	(1)
Method of solution	50	(2)
Solution at O(K)	52	(3)
Solution at O(ε)	55	(1)
Solution at O(K2)	56	(1)
Contribution of the O(K2) terms	57	(2)
Constitutive relations	59	(2)
Boundary Conditions	61	(11)
The elastic case	65	(3)
Solubility conditions and some results	68	(2)
The inelastic case	70	(2)
Conclusions, Problems and Outlook	72	(2)
References	74	(7)
Collective Behavior of One-Dimensional Granular Media	81	(30)
D. Benedetto
E. Caglioti
M. Pulvirenti
Introduction	81	(2)
The Microscopic Model	83	(1)
Collapses	84	(3)
The Quasielastic Limit	87	(3)
The Mean-Field Equation	90	(2)
The Hydrodynamic Behavior of the Mean Field Equation	92	(5)
One-Dimensional Boltzmann Equation	97	(3)
Heating the System	100	(3)
A Hydrodynamical Picture	103	(4)
The Diffusive Limit	107	(1)
References	108	(3)
Notes on Mathematical Problems on the Dynamics of Dispersed Particles Interacting through a Fluid	111	(38)
P.E. Jabin
B. Perthame
Introduction	111	(4)
Dynamics of Balls in a Potential Flow	115	(6)
The full dynamics	116	(2)
The method of reflections	118	(1)
The dipole approximation	119	(2)
Kinetic Theory for the Hamiltonian System of Bubbly Flows	121	(5)
The general Lagrangian structure	121	(1)
The corresponding Hamiltonian structure	122	(1)
The mean field equation	123	(3)
Numerical Simulation in the Case of a Potential Flow and Short Range Effect	126	(3)
Interaction of Particles in a Stokes Flow	129	(6)
Notations	130	(1)
Case of a single bubble and Stokeslets	131	(2)
The method of reflections	133	(1)
The dipole approximation	133	(2)
Kinetic and Macroscopic Equations for Particles in a Stokes Flow	135	(4)
The general interaction model	135	(2)
Energy and long time behavior for the kinetic equation	137	(1)
A macroscopic equation	138	(1)
Numerical Simulations for Stokes Flow	139	(6)
Introduction	139	(4)
Presentation of the computation	143	(1)
Conclusions	144	(1)
References	145	(4)
The Becker-Doring Equations	149	(24)
M. Slemrod
Introduction	149	(2)
Existence of Solutions to the Becker-Doring Equations	151	(4)
Trend to Equilibrium	155	(4)
Metastable States	159	(4)
Large Time Asymptotic Revised: Lifschitz-Slyozov and Wagner Evolution	163	(7)
References	170	(3)
Nonlinear Kinetic Models with Chemical Reactions	173	(52)
C.P. Grunfeld
Introduction	173	(5)
Boltzmann Equations for Reacting Gas	178	(14)
Extended kinetic theory with creation and removal	181	(2)
Generalized Boltzmann equations	183	(9)
General Properties of Solutions	192	(15)
The initial value problem	192	(5)
The H-theorem, equilibrium properties and mass action law	197	(5)
Outlines of proofs	202	(5)
Analytical Solutions, Approximation Methods, Reactive Fluid Dynamic Limits	207	(12)
Analytical solutions	208	(2)
Approximation methods	210	(4)
Reactive fluid dynamic limits	214	(5)
Concluding Remarks and Open Problems	219	(2)
References	221	(4)
Development of Boltzmann Models in Mathematical Biology	225	(38)
N. Bellomo
S. Stocker
Introduction	225	(2)
The Boltzmann Equation in Population Dynamics	227	(6)
A Few Notes on the Cauchy Problem	233	(1)
Application in Mathematical Epidemiology	234	(5)
Application in Mathematical Immunology	239	(9)
A Survey of Applications	248	(6)
Developments and Perspectives	254	(3)
Models with internal structure	254	(1)
Models with time structure	255	(1)
Research perspectives on modeling	256	(1)
Research perspectives on analytic topics	257	(1)
The Interplay between Mathematics and Immunology	257	(2)
References	259	(4)
Kinetic Traffic Flow Models	263	(54)
A. Klar
R. Wegener
Introduction	263	(1)
Basic Concepts	264	(6)
Levels of descriptions and notations	264	(2)
Homogeneous traffic flow	266	(4)
Microscopic Models	270	(6)
Car following models	270	(2)
A multilane microscopic model	272	(4)
Cellular automata models	276	(1)
Kinetic Models	276	(13)
The Prigogine model	277	(1)
The Paveri-Fontana model	278	(1)
Boltzmann versus Enskog type kinetic models	279	(1)
A kinetic multilane model	280	(9)
Macroscopic Models	289	(10)
Basic models	289	(1)
Models with an acceleration equation	290	(2)
A derived fluid dynamic model	292	(7)
Numerical Simulations	299	(14)
Simulation of the microscopic model	299	(4)
Simulation of the cumulative homogeneous kinetic model and computation of macroscopic coefficients	303	(4)
Inhomogeneous simulations	307	(6)
References	313	(4)
Kinetic Limits for Large Communication Networks	317	(54)
C. Grahm
Introduction	317	(4)
The scope of this document	319	(1)
Kinetic limits for chaotic initial laws and in equilibrium	319	(1)
Development of this document	320	(1)
Examples of Networks and of Related Practical Issues	321	(5)
Invariant laws, and the Erlag fixed point approximation	321	(1)
A star-shaped loss network	322	(2)
A queuing network with selection of the shortest among several queues	324	(1)
A fully-connected loss network with alternative routing	325	(1)
Preliminaries	326	(4)
General notation and terminology	326	(1)
The Skorohod space	327	(1)
General network notation	328	(1)
Chaoticity, exhangeability, and laws of large numbers	328	(2)
Mean-Field Networks and Propagation of Chaos	330	(5)
Mean-field models and nonlinear limits	330	(3)
Propagation of chaos	333	(2)
Chaoticity in Equilibrium	335	(3)
Chaoticity for the Star-Shaped Loss Network	338	(4)
Martingale formulations, and equations for the marginals	339	(2)
Propagation of chaos	341	(1)
Chaoticity in equilibrium	342	(1)
Chaoticity for the Queuing Network with Selection of the Shortest among Several Queues	342	(6)
Martingale formulations, and equations for the marginals	343	(1)
Propagation of chaos	344	(2)
Chaoticity in equilibrium	346	(2)
Propagation of Chaos Using Random Graphs and Trees	348	(11)
The fully connected loss network with alternative routing	348	(3)
Propagation of chaos for a general class of networks	351	(2)
The chaos hypothesis and the empirical measures	353	(4)
The limit Boltzmann tree and Boltzmann processes	357	(1)
Propagation of chaos under slight symmetry assumptions	358	(1)
Functional Central Limit and Large Deviation Results	359	(7)
Central limit theorems	359	(7)
Large deviation results	366	(1)
Conclusions and Perspectives	366	(2)
Propagation of chaos	366	(1)
Chaoticity in equilibrium	367	(1)
Central limit and large deviation results	367	(1)
References	368	(3)
Numerical Simulation of the Boltzmann Equation by Particle Methods	371
J. Struckmaier
Introduction	371
Particle Methods for the Boltzmann Equation	373
Approximation of functions by particles	374
Spatial-homogeneous Boltzmann equation	378
Spatial-inhomogeneous problems	383
Generalized time integration schemes	387
Extensions to steady-state problems	392
Numerical examples	395
Internal Degrees of Freedom and Chemical Reactions	399
The generalized Borgnakke-Larsen model	399
Extensions to chemically reacting flows	400
Numerical examples	402
Simulation Techniques on Parallel Computers	408
Simple parallel codes	408
Adaptive load balance techniques	409
References	415