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Zusatztext The authors introduce a new class of structure preserving numerical methods which improve the qualitative behavior of solutions of partial differential equations and allow stable computing. ? This book should be useful to engineers and physicists with a basic knowledge of numerical analysis.-Rémi Vaillancourt! Mathematical Reviews! Issue 2011m Informationen zum Autor Daisuke Furihata, Takayasu Matsuo Klappentext Nonlinear Partial Differential Equations (PDEs) have become increasingly important in the description of physical phenomena. Unlike Ordinary Differential Equations, PDEs can be used to effectively model multidimensional systems. The methods put forward in this book concentrate on a new class of "structure-preserving numerical equations" which improves the qualitative behaviour of the PDE solutions and allows for stable computing. The authors have also taken care to present their methods in an accessible manner, which means that the book will be useful to engineers and physicists with a basic knowledge of numerical analysis. Zusammenfassung Many important problems in engineering and science are modeled by nonlinear partial differential equations (PDEs). A new trend in PDEs, called structure-preserving numerical methods, has recently developed. This book introduces key factors and basic ideas of discrete variational derivative method. Inhaltsverzeichnis PrefaceIntroduction and Summary of This Book An Introductory Example: the Spinodal DecompositionHistoryDerivation of Dissipative or Conservative SchemesAdvanced TopicsTarget Partial Differential Equations Variational DerivativesFirst-Order Real-Valued PDEsFirst-Order Complex-Valued PDEsSystems of First-Order PDEsSecond-Order PDEsDiscrete Variational Derivative Method Discrete Symbols and FormulasProcedure for First-Order Real-Valued PDEsProcedure for First-Order Complex-Valued PDEsProcedure for Systems of First-Order PDEsDesign of SchemesProcedure for Second-Order PDEsPreliminaries on Discrete Functional AnalysisApplications Target PDEs Cahn-Hilliard EquationAllen-Cahn EquationFisher-Kolmogorov EquationTarget PDEs Target PDEs Target PDEs Nonlinear Schr¨odinger EquationTarget PDEs Zakharov EquationsTarget PDEs Other EquationsAdvanced Topic I: Design of High-Order Schemes Orders of Accuracy of the SchemesSpatially High-Order SchemesTemporally High-Order Schemes: With the Composition MethodTemporally High-Order Schemes: With High-Order Discrete Variational DerivativesAdvanced Topic II: Design of Linearly-Implicit Schemes Basic Idea for Constructing Linearly-Implicit SchemesMultiple-Points Discrete Variational DerivativeDesign of SchemesApplicationsRemark on the Stability of Linearly-Implicit SchemesAdvanced Topic III: Further Remarks Solving System of Nonlinear EquationsSwitch to Galerkin FrameworkExtension to Non-Rectangular Meshes on D Region A Semi-discrete schemes in spaceB Proof of Proposition 3.4BibliographyIndex ...
