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Chapter 7. Numerical Solutions Using Finite Element Methods
	7.1 Related Principles in Variational Calculus
	7.1.1 Partial Differential Equations and Their Variational Formulation
	7.1.2 Condition of Existence of Variational Formulation
	7.1.3 Approximate Solution of Differential Equations with the Rayleigh–Ritz Method
	7.1.4 Approximate Solution of Differential Equation with Weighted Residual Method
	7.1.5 Further Discussions on the Galerkin Method
	7.2 Piecewise Approximation of Plane Problems and Convergence of FEM Solutions
	7.2.1 Basic Requirements for Element Partitioning and Piecewise Approximation
	7.2.2 Element Partitioning and Interpolation for Plane Problems
	7.2.3 Theoretical Analysis of FEM
	7.3 FEM for 2-D Unsteady Open Flows
	7.3.1 Derivation of Galerkin Equations
	7.3.2 Calculation of Influence Coefficients and Related Integrals for Triangular Elements
	7.4 Several Classes of Special FEMs
	7.4.1 Nonconforming Element and Hybrid Element
	7.4.2 Mixed Element and Mixed Interpolation
	7.4.3 Explicit FEM
	7.4.4 Upstream FEM and Petrov–Galerkin FEM
	Chapter 8. Techniques for the Implementation of Algorithms
	8.1 Computational Mesh
	8.1.1 Introduction
	8.1.2 Computational Domain and Transformed Domain
	8.1.3 General Principles for Numerical Generation of a Curvilinear Mesh
	8.1.4 Numerical Generation of a Boundary-Fitted Orthogonal Curvilinear Mesh
	8.1.5 Staggered Mesh for 2-D SSWE
	8.1.6 Mesh with Nonuniform Step Size
	8.1.7 Nested Mesh
	8.1.8 Movable Boundary
	8.1.9 Automatic Mesh Generation
	8.1.10 Irregular Mesh, Lagrangian Mesh, and Adaptive Mesh
	8.2 Classical Techniques for Improving Computational Stability and Accuracy
	8.2.1 General Description
	8.2.2 Numerical Filtering
	8.2.3 Two Classes of Techniques for Dealing with Discontinuous Solutions
	8.2.4 Artificial Viscosity Method
	8.2.5 Comparison between Chief Classes of Difference Schemes
	8.2.6 Richardson Extrapolation
	8.2.7 Time-Integration Schemes and Time-Correlation Method
	Chapter 9. New Developments of Difference Schemes for 2-D First-Order Hyperbolic Systems of Equations
	9.1 General Description
	9.1.1 Review of Historical Development
	9.1.2 Construction of Upwind Explicit Difference Schemes in Conservative Form
	9.1.3 Convergence of a Numerical Solution to a Physical Solution
	9.2 Two-Dimensional Methods of Characteristics
	9.2.1 General Description
	9.2.2 Moretti Lambda Scheme
	9.3 Characteristic-Based Splitting
	9.3.1 Flux Vector Splitting (FVS)
	9.3.2 Applications of FVS to the Solution of SSWE
	9.3.3 Flux-Difference Splitting (FDS) and Its Application
	9.3.4 Osher–Solomon Scheme and Its Application to the Solution of 2-D SSWE
	9.4 Riemann Approach
	9.4.1 Solution of 1-D Riemann Problems
	9.4.2 Godunov and Godunov-Type Scheme
	9.4.3 Glimm Scheme
	9.5 Approximate Factorization of Implicit Schemes
	9.6 FCT Algorithms and TVD Schemes
	9.6.1 Flux-Corrected Transport (FCT) Algorithms
	9.6.2 Total Variation Diminishing (TVD) Schemes
	9.7 Square Conservation Schemes
	9.7.1 General Description
	9.7.2 Construction of a Square Conservation Scheme
	Chapter 10. Stability Analysis and Boundary Procedures
	10.1 Mathematical Definitions of Stability for Difference Schemes
	10.1.1 The First Definition
	10.1.2 The Second Definition
	10.2 Von Neumann Linear Stability Analysis
	10.2.1 Von Neumann Method
	10.2.2 Discussions on the Stability Condition
	10.3 Nonlinear Instability
	10.3.1 Introduction
	10.3.2 Energy Method
	10.3.3 Hirt Heuristic Method and Approximate Differential Equations
	10.4 Boundary Procedures and Their Influence on Numerical Solutions
	10.4.1 General Description
	10.4.2 Schemes Used for Boundary Nodes
	10.4.3 Influence of Boundary Procedures on Numerical Solutions
	10.5 Stability Theory for Mixed Problems
	10.5.1 Description of the Problem
	10.5.2 Necessary Condition of Stability
	10.5.3 Outline of the GKS Theory
	10.5.4 Sufficient Condition of Stability
	Chapter 11. Concluding Remarks
	11.1 Requirements for an Ideal Finite-Difference Scheme
	11.2 Comparison of Performance, Merits and Drawbacks between FDM and FEM
	11.3 Brief Introduction to Other Algorithms
	11.3.1 Spectral Method
	11.3.2 Methods of Lines
	11.3.3 Finite Analytic Method
	11.3.4 Lagrangian Approach
	11.3.5 Boundary Element Method (Boundary Integration Method)
	11.4 Towards a Truly 2-D Algorithm