Finite Difference Methods For Nonlinear Evolution Equations

Introduces recent research results of finite difference methods including important nonlinear evolution equations in applied science. The presented difference schemes include nonlinear difference schemes and linearized difference schemes. Features widely used nonlinear evolution equations such as Burgers equation, regular long wave equation, Schrodinger equation and more. Each PDE model includes details on efficiency, stability, and convergence.

This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Nonlinear evolution equations are widely used to describe nonlinear phenomena in natural and social sciences. However, they are usually quite difficult to solve in most instances. This book introduces the finite difference methods for solving nonlinear evolution equations. The main numerical analysis tool is the energy method. This book covers the difference methods for the initial-boundary value problems of twelve nonlinear partial differential equations. They are Fisher equation, Burgers' equation, regularized long-wave equation, Korteweg-de Vries equation, Camassa-Holm equation, Schrödinger equation, Kuramoto-Tsuzuki equation, Zakharov equation, Ginzburg-Landau equation, Cahn-Hilliard equation, epitaxial growth model and phase field crystal model. This book is a monograph for the graduate students and science researchers majoring in computational mathematics and applied mathematics. It will be also useful to all researchers in related disciplines.

The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter I gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations, that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used. This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.

This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances whereas in using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of “exact” and “best” finite difference schemes. In addition, a set of rules is given for the discrete modeling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique nonstandard finite difference model for a given differential equation. Contents:IntroductionNumerical InstabilitiesNonstandard Finite-Difference SchemesFirst-Order ODE'sSecond-Order, Nonlinear Oscillator EquationsTwo First-Order, Coupled Ordinary Differential EquationsPartial Differential EquationsSchrädinger Differential EquationsSummary and DiscussionAppendices: Difference EquationsLinear Stability AnalysisDiscrete WKB MethodBibliographyIndex Readership: Applied mathematicians (numerical analysis and modeling). keywords:Finite Difference Techniques;Numerical Schemes for Differential Equations;Numerical Instabilities;Nonstandard Schemes;Exact Finite Difference Schemes;Best Finite Difference Schemes;Denominator Functions;Linear Stability Analysis;Discrete WKB Method “This book contains a clear presentation of nonstandard finite difference schemes for the numerical integration of differential equations. A set of rules for constructing nonstandard finite difference schemes is also presented. An important feature of the book is the illustration of the various discrete modeling principles, by their application to a large number of both ordinary and partial differential equations.” Mathematical Reviews

What makes this book stand out from the competition is that it is more computational. Once done with both volumes, readers will have the tools to attack a wider variety of problems than those worked out in the competitors' books. The author stresses the use of technology throughout the text, allowing students to utilize it as much as possible.

This book presents an approximation theory for a general class of nonlinear evolution equations in Banach spaces and the semigroup theory, including the linear (Hille–Yosida), nonlinear (Crandall–Liggett) and time-dependent (Crandall–Pazy) theorems. The implicit finite difference method of Euler is shown to generate a sequence convergent to the unique integral solution of evolution equations of the maximal monotone type. Moreover, the Chernoff theory provides a sufficient condition for consistent and stable time integration of time-dependent nonlinear equations. The Trotter–Kato theorem and the Lie–Trotter type product formula give a mathematical framework for the convergence analysis of numerical approximations of solutions to a general class of partial differential equations. This book contains examples demonstrating the applicability of the generation as well as the approximation theory. In addition, the Kobayashi–Oharu approach of locally quasi-dissipative operators is discussed for homogeneous as well as nonhomogeneous equations. Applications to the delay differential equations, Navier–Stokes equation and scalar conservation equation are given. Contents: Dissipative and Maximal Monotone OperatorsLinear SemigroupsAnalytic SemigroupsApproximation of C0-SemigroupsNonlinear Semigroups of ContractionsLocally Quasi-Dissipative Evolution EquationsThe Crandall–Pazy ClassVariational Formulations and Gelfand TriplesApplications to Concrete SystemsApproximation of Solutions for Evolution EquationsSemilinear Evolution EquationsAppendices:Some InequalitiesConvergence of Steklov MeansSome Technical Results Needed in Section 9.2 Readership: Researchers in the fields of analysis & differential equations and approximation theory. Keywords:Evolution Equations;Approximations;Euler;Trotter-Kato;Lie-Trotter;Quasi-Dissipative Operators;K and Y Kobayashi;S OharuReviews:“Ito and Kappel offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K and Y Kobayashi and S Oharu … their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses.”Book News, Inc.

This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered.