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Introduction | |

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Motivation | |

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Problem Specification and Geometry Preparation | |

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Selection of Governing Equations and Boundary Conditions | |

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Selection of Gridding Strategy and Numerical Method | |

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Assessment and Interpretation of Results | |

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Overview | |

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Notation | |

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Conservation Laws and the Model Equations | |

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Conservation Laws | |

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The Navier-Stokes and Euler Equations | |

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The Linear Convection Equation | |

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Differential Form | |

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Solution in Wave Space | |

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The Diffusion Equation | |

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Differential Form | |

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SolutioninWaveSpace | |

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Linear Hyperbolic Systems | |

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Exercises | |

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Finite-Difference Approximations | |

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Meshes and Finite-Difference Notation | |

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Space DerivativeApproximations | |

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Finite-Difference Operators | |

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Point Difference Operators | |

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Matrix Difference Operators | |

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Periodic Matrices | |

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CirculantMatrices | |

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Constructing Differencing Schemes of Any Order | |

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TaylorTables | |

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Generalization of Difference Formulas | |

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Lagrange and Hermite Interpolation Polynomials | |

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Practical Application of Padï¿½ Formulas | |

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OtherHigher-OrderSchemes | |

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FourierErrorAnalysis | |

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ApplicationtoaSpatialOperator | |

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Difference Operators at Boundaries | |

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TheLinearConvectionEquation | |

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The Diffusion Equation | |

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Exercises | |

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The Semi-Discrete Approach | |

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Reduction of PDE's to ODE's | |

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The Model ODE's | |

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TheGenericMatrixForm | |

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ExactSolutionsofLinearODE's | |

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EigensystemsofSemi-discreteLinearForms | |

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Single ODE's of First and Second Order | |

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CoupledFirst-OrderODE's | |

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General Solution of Coupled ODE's with Complete Eigensystems | |

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RealSpaceandEigenspace | |

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Definition | |

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EigenvalueSpectrumsforModelODE's | |

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Eigenvectors of the Model Equations | |

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Solutions of the Model ODE's | |

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TheRepresentative Equation | |

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Exercises | |

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Finite-Volume Methods | |

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Basic Concepts | |

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ModelEquations in Integral Form | |

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TheLinearConvectionEquation | |

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The Diffusion Equation | |

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One-DimensionalExamples | |

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A Second-Order Approximation to the Convection Equation | |

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A Fourth-Order Approximation to the Convection Equation | |

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A Second-Order Approximation to the Diffusion Equation | |

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ATwo-Dimensional Example | |

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Exercises | |

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Time-Marching Methods for ODE'S | |

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Notation | |

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Converting Time-Marching Methods to Oï¿½E's | |

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Solution of Linear Oï¿½E's with Constant Coefficients | |

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First- and Second-Order Difference Equations | |

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Special Cases of Coupled First-Order Equations | |

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Solution of the Representative Oï¿½E's | |

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The Operational Form and its Solution | |

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Examples of Solutions to Time-Marching Oï¿½E's | |

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The ï¿½-ï¿½ Relation | |

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Establishing the Relation | |

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The Principal ï¿½-Root | |

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Spurious ï¿½-Roots | |

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One-Root Time-Marching Methods | |

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Accuracy Measures of Time-Marching Methods | |

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Local and Global Error Measures | |

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Local Accuracy of the Transient Solution (er<sub>ï¿½</sub>ï¿½, er<sub>&omgea;) | |

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Local Accuracy of the Particular Solution (er<sub>ï¿½</sub>) | |

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Time Accuracy for Nonlinear Applications | |

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Global Accuracy | |

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Linear Multistep Methods | |

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The General Formulation | |

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Examples | |

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Two-Step Linear Multistep Methods | |

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Predictor-Corrector Methods | |

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Runge-Kutta Methods | |

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Implementation of Implicit Methods | |

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Application to Systems of Equations | |

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Application to Nonlinear Equations | |

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Local Linearization for Scalar Equations | |

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Local Linearization for Coupled Sets of Nonlinear Equations | |

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Exercises | |

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Stability of Linear Systems | |

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Dependence on the Eigensystem | |

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Inherent Stability of ODE's | |

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The Criterion | |

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Complete Eigensystems | |

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Defective Eigensystems | |

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Numerical Stability of Oï¿½E's | |

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The Criterion | |

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Complete Eigensystems | |

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Defective Eigensystems | |

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Time-Space Stability and Convergence of Oï¿½E's | |

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Numerical Stability Concepts in the Complex ï¿½-Plane | |

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ï¿½-Root Traces Relative to the Unit Circle | |

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Stability for Small ï¿½t | |

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Numerical Stability Concepts in the Complex ï¿½h Plane | |

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Stability for Large h | |

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Unconditional Stability, A-Stable Methods | |

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Stability Contours in the Complex ï¿½h Plane | |

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Fourier Stability Analysis | |

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The Basic Procedure | |

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Some Examples | |

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Relation to Circulant Matrices | |

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Consistency | |

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Exercises | |

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Choosing a Time-Marching Method | |

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Stiffness Definition for ODE's | |

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Relation to ï¿½-Eigenvalues | |

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Driving and Parasitic Eigenvalues | |

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Stiffness Classifications | |

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Relation of Stiffness to Space Mesh Size | |

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Practical Considerations for Comparing Methods | |

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Comparing the Efficiency of Explicit Methods | |

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Imposed Constraints | |

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An Example Involving Diffusion | |

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An Example Involving Periodic Convection | |

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Coping with Stiffness | |

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Explicit Methods | |

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Implicit Methods | |

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A Perspective | |

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Steady Problems | |

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Exercises | |

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Relaxation Methods | |

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Formulation of the Model Problem | |

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Preconditioning the Basic Matrix | |

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The Model Equations | |

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Classical Relaxation | |

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The Delta Form of an Iterative Scheme | |

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The Converged Solution, the Residual, and the Error | |

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The Classical Methods | |

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The ODE Approach to Classical Relaxation | |

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The Ordinary Differential Equation Formulation | |

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ODE Form of the Classical Methods | |

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Eigensystems of the Classical Methods | |

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The Point-Jacobi System | |

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The Gauss-Seidel System | |

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The SOR System | |

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Nonstationary Processes | |

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Exercises | |

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Multigrid | |

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Motivation | |

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Eigenvector and Eigenvalue Identification with Space Frequencies | |

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Properties of the Iterative Method | |

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The Basic Process | |

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A Two-Grid Process | |

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Exercises | |

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Numerical Dissipation | |

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One-Sided First-Derivative Space Differencing | |

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The Modified Partial Differential Equation | |

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The Lax-Wendroff Method | |

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Upwind Schemes | |

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Flux-Vector Splitting | |

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Flux-Difference Splitting | |

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Artificial Dissipation | |

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Exercises | |

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Split and Factored Forms | |

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The Concept | |

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Factoring Physical Representations -Time Splitting | |

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Factoring Space MatrixOperators in 2D | |

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Mesh Indexing Convention | |

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Data-Bases and Space Vectors | |

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Data-Base Permutations | |

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Space Splitting and Factoring | |

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Second-Order Factored Implicit Methods | |

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Importance of Factored Forms in Two and Three Dimensions | |

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The Delta Form | |

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Exercises | |

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Analysis of Split and Factored Forms | |

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The Representative Equation for Circulant Operators | |

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Example Analysis of Circulant Systems | |

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Stability Comparisons of Time-Split Methods | |

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Analysisofa Second-Order Time-Split Method | |

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The Representative Equation for Space-Split Operators | |

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Example Analysis of the 2D Model Equation | |

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The Unfactored Implicit Euler Method | |

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The Factored Nondelta Form of the Implicit Euler Method | |

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The Factored Delta Form of the Implicit Euler Method | |

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The Factored Delta Form of the Trapezoidal Method | |

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Example Analysis of the 3D Model Equation | |

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Exercises | |

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Appendices | |

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Useful Relations from Linear Algebra | |

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Notation | |

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Definitions | |

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Algebra | |

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Eigensystems | |

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Vector and Matrix Norms | |

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Some Properties of Tridiagonal Matrices | |

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Standard Eigensystem for Simple Tridiagonal Matrices | |

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Generalized Eigensystem for Simple Tridiagonal Matrices | |

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The Inverse of a Simple Tridiagonal Matrix | |

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Eigensystems of Circulant Matrices | |

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Standard Tridiagonal Matrices | |

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General Circulant Systems | |

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Special Cases Found from Symmetries | |

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Special Cases Involving Boundary Conditions | |

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The Homogeneous Property of the Euler Equations | |

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Index | |