Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations
Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations
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DOI：
10.1201/b10387
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发表时间：
2010-12
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影响因子：
--
通讯作者：
Daisuke Furihata;Takayasu Matsuo
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文献类型：
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作者：
Daisuke Furihata;Takayasu Matsuo
前言本书的介绍和总结一个介绍性的例子：Spinodal分解历史耗散或守恒格式的推导高级主题目标偏微分方程变分导数一阶实值偏微分方程一阶复值偏微分方程一阶偏微分方程系统二阶偏微分方程离散变分导数方法离散符号和公式一阶实值偏微分方程程序一阶复偏微分方程程序一阶偏微分方程系统的数值偏微分方程程序方案设计二阶偏微分方程程序离散泛函分析应用程序目标偏微分方程Cahn-Hilliard方程Allen-Cahn方程Fisher-Kolmogorov方程目标偏微分方程非线性薛定谔方程目标偏微分方程Zakharov方程目标偏微分方程其他方程高级主题I：高阶格式的设计格式的精度阶数空间高阶格式时间高阶格式：采用合成法时间高阶格式：采用高阶离散变分导数高级专题II：线性隐式格式的设计构造线性隐式格式的基本思想多点离散变分导数格式的设计应用线性隐式格式稳定性的注记进一步注记求解非线性方程组转换到Galerkin框架D区域上非矩形网格的推广A半空间离散格式B命题3.4的证明
Preface Introduction and Summary of This Book An Introductory Example: the Spinodal Decomposition History Derivation of Dissipative or Conservative Schemes Advanced Topics Target Partial Differential Equations Variational Derivatives First-Order Real-Valued PDEs First-Order Complex-Valued PDEs Systems of First-Order PDEs Second-Order PDEs Discrete Variational Derivative Method Discrete Symbols and Formulas Procedure for First-Order Real-Valued PDEs Procedure for First-Order Complex-Valued PDEs Procedure for Systems of First-Order PDEs Design of Schemes Procedure for Second-Order PDEs Preliminaries on Discrete Functional Analysis Applications Target PDEs Cahn-Hilliard Equation Allen-Cahn Equation Fisher-Kolmogorov Equation Target PDEs Target PDEs Target PDEs Nonlinear Schrodinger Equation Target PDEs Zakharov Equations Target PDEs Other Equations Advanced Topic I: Design of High-Order Schemes Orders of Accuracy of the Schemes Spatially High-Order Schemes Temporally High-Order Schemes: With the Composition Method Temporally High-Order Schemes: With High-Order Discrete Variational Derivatives Advanced Topic II: Design of Linearly-Implicit Schemes Basic Idea for Constructing Linearly-Implicit Schemes Multiple-Points Discrete Variational Derivative Design of Schemes Applications Remark on the Stability of Linearly-Implicit Schemes Advanced Topic III: Further Remarks Solving System of Nonlinear Equations Switch to Galerkin Framework Extension to Non-Rectangular Meshes on D Region A Semi-discrete schemes in space B Proof of Proposition 3.4 Bibliography Index