Fundamental Concepts / Chapter 1.: |
Notation / 1.0.: |
First-Order Partial Differential Equations / 1.1.: |
First-Order Quasilinear Partial Differential Equations / 1.1.1.: |
Initial Value or Cauchy Problem / 1.1.2.: |
Application of Characteristic Curves / 1.1.3.: |
Nonlinear First-Order Partial Differential Equations / 1.1.4.: |
Second-Order Partial Differential Equations / 1.2.: |
Linear Second-Order Partial Differential Equations / 1.2.1.: |
Classification and Canonical Form of Selected Partial Differential Equations / 1.2.2.: |
Quasilinear Partial Differential Equations and Other Ideas / 1.2.3.: |
Systems of First-Order PDEs / 1.3.: |
First-Order and Second-Order PDEs / 1.3.1.: |
Characteristic Curves / 1.3.2.: |
Applications of Characteristic Curves / 1.3.3.: |
Initial and Boundary Conditions / 1.4.: |
References |
Basic Concepts in the Finite Difference and Finite Element Methods / Chapter 2.: |
Introduction / 2.0.: |
Finite Difference Approximations / 2.1.: |
Taylor Series Expansions / 2.1.1.: |
Operator Notation for u(x) / 2.1.3.: |
Finite Difference Approximations in Two Dimensions / 2.1.4.: |
Additional Concepts / 2.1.5.: |
Introduction to Finite Element Approximations / 2.2.: |
Method of Weighted Residuals / 2.2.1.: |
Application of the Method of Weighted Residuals / 2.2.2.: |
The Choice of Basis Functions / 2.2.3.: |
Two-Dimensional Basis Functions / 2.2.4.: |
Approximating Equations / 2.2.5.: |
Relationship between Finite Element and Finite Difference Methods / 2.3.: |
Finite Elements on Irregular Subspaces / Chapter 3.: |
Triangular Elements / 3.0.: |
The Linear Triangular Element / 3.1.1.: |
Area Coordinates / 3.1.2.: |
The Quadratic Triangular Element / 3.1.3.: |
The Cubic Triangular Element / 3.1.4.: |
Higher-Order Triangular Elements / 3.1.5.: |
Isoparametric Finite Elements / 3.2.: |
Transformation Functions / 3.2.1.: |
Numerical Integration / 3.2.2.: |
Isoparametric Serendipity Hermitian Elements / 3.2.3.: |
Isoparametric Hermitian Elements in Normal and Tangential Coordinates / 3.2.4.: |
Boundary Conditions / 3.3.: |
Three-Dimensional Elements / 3.4.: |
Parabolic Partial Differential Equations / Chapter 4.: |
Partial Differential Equations / 4.0.: |
Well-Posed Partial Differential Equations / 4.1.1.: |
Model Difference Approximations / 4.2.: |
Well-Posed Difference Forms / 4.2.1.: |
Derivation of Finite Difference Approximations / 4.3.: |
The Classic Explicit Approximation / 4.3.1.: |
The Dufort-Frankel Explicit Approximation / 4.3.2.: |
The Richardson Explicit Approximation / 4.3.3.: |
The Backwards Implicit Approximation / 4.3.4.: |
The Crank-Nicolson Implicit Approximation / 4.3.5.: |
The Variable-Weighted Implicit Approximation / 4.3.6.: |
Consistency and Convergence / 4.4.: |
Stability / 4.5.: |
Heuristic Stability / 4.5.1.: |
Von Neumann Stability / 4.5.2.: |
Matrix Stability / 4.5.3.: |
Some Extensions / 4.6.: |
Influence of Lower-Order Terms / 4.6.1.: |
Higher-Order Forms / 4.6.2.: |
Predictor-Corrector Methods / 4.6.3.: |
Asymmetric Approximations / 4.6.4.: |
Variable Coefficients / 4.6.5.: |
Nonlinear Parabolic PDEs / 4.6.6.: |
The Box Method / 4.6.7.: |
Solution of Finite Difference Approximations / 4.7.: |
Solution of Implicit Approximations / 4.7.1.: |
Explicit versus Implicit Approximations / 4.7.2.: |
Composite Solutions / 4.8.: |
Global Extrapolation / 4.8.1.: |
Some Numerical Results / 4.8.2.: |
Local Combination / 4.8.3.: |
Composites of Different Approximations / 4.8.4.: |
Finite Difference Approximations in Two Space Dimensions / 4.9.: |
Explicit Methods / 4.9.1.: |
Irregular Boundaries / 4.9.2.: |
Implicit Methods / 4.9.3.: |
Alternating Direction Explicit (ADE) Methods / 4.9.4.: |
Alternating Direction Implicit (ADI) Methods / 4.9.5.: |
LOD and Fractional Splitting Methods / 4.9.6.: |
Hopscotch Methods / 4.9.7.: |
Mesh Refinement / 4.9.8.: |
Three-Dimensional Problems / 4.10.: |
ADI Methods / 4.10.1.: |
Iterative Solutions / 4.10.2.: |
Finite Element Solution of Parabolic Partial Differential Equations / 4.11.: |
Galerkin Approximation to the Model Parabolic Partial Differential Equation / 4.11.1.: |
Approximation of the Time Derivative / 4.11.2.: |
Approximation of the Time Derivative for Weakly Nonlinear Equations / 4.11.3.: |
Finite Element Approximations in One Space Dimension / 4.12.: |
Formulation of the Galerkin Approximating Equations / 4.12.1.: |
Linear Basis Function Approximation / 4.12.2.: |
Higher-Degree Polynomial Basis Function Approximation / 4.12.3.: |
Formulation Using the Dirac Delta Function / 4.12.4.: |
Orthogonal Collocation Formulation / 4.12.5.: |
Asymmetric Weighting Functions / 4.12.6.: |
Finite Element Approximations in Two Space Dimensions / 4.13.: |
Galerkin Approximation in Space and Time / 4.13.1.: |
Galerkin Approximation in Space Finite Difference in Time / 4.13.2.: |
Asymmetric Weighting Functions in Two Space Dimensions / 4.13.3.: |
Lumped and Consistent Time Matrices / 4.13.4.: |
Collocation Finite Element Formulation / 4.13.5.: |
Treatment of Sources and Sinks / 4.13.6.: |
Alternating Direction Formulation / 4.13.7.: |
Finite Element Approximations in Three Space Dimensions / 4.14.: |
Example Problem / 4.14.1.: |
Summary / 4.15.: |
Elliptic Partial Differential Equations / Chapter 5.: |
Model Elliptic PDEs / 5.0.: |
Specific Elliptic PDEs / 5.1.1.: |
Further Items / 5.1.2.: |
Finite Difference Solutions in Two Space Dimensions / 5.2.: |
Five-Point Approximations and Truncation Error / 5.2.1.: |
Nine-Point Approximations and Truncation Error / 5.2.2.: |
Approximations to the Biharmonic Equation / 5.2.3.: |
Boundary Condition Approximations / 5.2.4.: |
Matrix Form of Finite Difference Equations / 5.2.5.: |
Direct Methods of Solution / 5.2.6.: |
Iterative Concepts / 5.2.7.: |
Formulation of Point Iterative Methods / 5.2.8.: |
Convergence of Point Iterative Methods / 5.2.9.: |
Line and Block Iteration Methods / 5.2.10.: |
Acceleration and Semi-Iterative Overlays / 5.2.11.: |
Finite Difference Solutions in Three Space Dimensions / 5.3.: |
Iteration Concepts / 5.3.1.: |
Finite Element Methods for Two Space Dimensions / 5.3.3.: |
Galerkin Approximation / 5.4.1.: |
Collocation Approximation / 5.4.2.: |
Mixed Finite Element Approximation / 5.4.4.: |
Approximation of the Biharmonic Equation / 5.4.5.: |
Boundary Integral Equation Methods / 5.5.: |
Fundamental Theory / 5.5.1.: |
Boundary Element Formulation / 5.5.2.: |
Linear Interpolation Functions / 5.5.3.: |
Poisson's Equation / 5.5.5.: |
Nonhomogeneous Materials / 5.5.6.: |
Combination of Finite Element and Boundary Integral Equation Methods / 5.5.7.: |
Three-Dimensional Finite Element Simulation / 5.6.: |
Hyperbolic Partial Differential Equations / 5.7.: |
Equations of Hyperbolic Type / 6.0.: |
Finite Difference Solution of First-Order Scalar Hyperbolic Partial Differential Equations / 6.2.: |
Stability, Truncation Error, and Other Features / 6.2.1.: |
Other Approximations / 6.2.2.: |
Dissipation and Dispersion / 6.2.3.: |
Hopscotch Methods and Mesh Refinement / 6.2.4.: |
Finite Difference Solution of First-Order Vector Hyperbolic Partial Differential Equations / 6.3.: |
Finite Difference Solution of First-Order Vector Conservative Hyperbolic Partial Differential Equations / 6.4.: |
Finite Difference Solutions to Two- and Three-Dimensional Hyperbolic Partial Differential Equations / 6.5.: |
Finite Difference Schemes / 6.5.1.: |
Two-Step, ADI, and Strang-Type Algorithms / 6.5.2.: |
Conservative Hyperbolic Partial Differential Equations / 6.5.3.: |
Finite Difference Solution of Second-Order Model Hyperbolic Partial Differential Equations / 6.6.: |
One-Space-Dimension Hyperbolic Partial Differential Equation / 6.6.1.: |
Explicit Algorithms / 6.6.2.: |
Implicit Algorithms / 6.6.3.: |
Simultaneous First-Order Partial Differential Equations / 6.6.4.: |
Mixed Systems / 6.6.5.: |
Two- and Three-Space-Dimensional Hyperbolic Partial Differential Equations / 6.6.6.: |
Implicit ADI and LOD Methods / 6.6.7.: |
Finite Element Solution of First-Order Model Hyperbolic Partial Differential Equations / 6.7.: |
Asymmetric Weighting Function Approximation / 6.7.1.: |
An H[superscript -1] Galerkin Approximation / 6.7.3.: |
Orthogonal Collocation with Asymmetric Bases / 6.7.4.: |
Finite Element Solution of Two- and Three-Space-Dimensional First-Order Hyperbolic Partial Differential Equations / 6.7.6.: |
Galerkin Finite Element Formulation / 6.8.1.: |
Finite Element Solution of First-Order Vector Hyperbolic Partial Differential Equations / 6.8.2.: |
Finite Element Solution of Two- and Three-Space-Dimensional First-Order Vector Hyperbolic Partial Differential Equations / 6.9.1.: |
Finite Element Solution of One-Space-Dimensional Second-Order Hyperbolic Partial Differential Equations / 6.10.1.: |
Time Approximations / 6.11.1.: |
Finite Element Solution of Two- and Three-Space-Dimensional Second-Order Hyperbolic Partial Differential Equations / 6.11.3.: |
Index / 6.12.1.: |
Fundamental Concepts / Chapter 1.: |
Notation / 1.0.: |
First-Order Partial Differential Equations / 1.1.: |