Finite Difference Methods for Nonlinear Evolution Equations

The series is devoted to the publication of high-level monographs and specialized graduate texts which cover the whole spectrum of applied mathematics, including its numerical aspects. The focus of the series is on the interplay between mathematical and numerical analysis, and also on its applications to mathematical models in the physical and life sciences. The aim of the series is to be an active forum for the dissemination of up-to-date information in the form of authoritative works that will serve the applied mathematics community as the basis for further research. Editorial Board Rémi Abgrall, Universität Zürich, Switzerland José Antonio Carrillo de la Plata, University of Oxford, UK Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France Athanassios S. Fokas, Cambridge University, UK Irene Fonseca, Carnegie Mellon University, Pittsburgh, USA 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. Preface About the Authors Contents 1 Difference methods for the Fisher equation 2 Difference methods for the Burgers’ equation 3 Difference methods for the regularized long-wave equation 4 Difference methods for the Korteweg–de Vries equation 5 Difference methods for the Camassa–Holm equation 6 Difference methods for the Schrödinger equation 7 Difference methods for the Kuramoto–Tsuzuki equation 8 Difference methods for the Zakharov equation 9 Difference methods for the Ginzburg–Landau equation 10 Difference methods for the Cahn–Hilliard equation 11 Difference methods for the epitaxial growth model 12 Difference methods for the phase field crystal model Bibliography Index