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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 "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 is devoted to one such technique! called the discrete variational derivative method. First! the text introduces the key factors and the basic ideas of this method! followed by target problems solvable by the method. The second section describes the rigorous mathematics in detail along with relevant applications! which are illustrated by worked examples. It concludes with a comprehensive of listing of essential references on structure-preserving algorithms for advanced readers"-- 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 ...
