Introductory Finite Difference Methods for PDEs
Publisher: Bookboon
Published: 2013
ISBN-13: 9788776816421
Table of contents: title: Introduction; title: Partial Differential Equations; title: Solution to a Partial Differential Equation; title: PDE Models; title: Classification of PDEs; title: Discrete Notation; title: Checking Results; title: Exercise 1; title: Fundamentals; title: Taylor’s Theorem; title: Taylor’s Theorem Applied to the Finite Difference Method (FDM); title: Simple Finite Difference Approximation to a Derivative; title: Example: Simple Finite Difference Approximations to a Derivative; title: Constructing a Finite Difference Toolkit; title: Simple Example of a Finite Difference Scheme; title: Pen and Paper Calculation (very important); title: Exercise 2a; title: Exercise 2b; title: Elliptic Equations; title: Introduction; title: Finite Difference Method for Laplace’s Equation; title: Setting up the Equations; title: Grid Convergence; title: Direct Solution Method; title: Exercise 3a; title: Iterative Solution Methods; title: Jacobi Iteration; title: Gauss-Seidel Iteration; title: Exercise 3b; title: Successive Over Relaxation (SoR) Method; title: Line SoR; title: Exercise 3c; title: Hyperbolic Equations; title: Introduction; title: 1D Linear Advection Equation; title: Results for the Simple Linear Advection Scheme; title: Scheme Design; title: Multi-Level Scheme Design; title: Exercise 4a; title: Implicit Schemes; title: Exercise 4b; title: Parabolic Equations: the Advection-Diffusion Equation; title: Introduction; title: Pure Diffusion; title: Advection-Diffusion Equation; title: Exercise 5b; title: Extension to Multi-dimensions and Operator Splitting; title: Introduction; title: 2D Scheme Design (unsplit); title: Operator Splitting (Approximate Factorisation); title: Systems of Equations; title: Introduction; title: The Shallow Water Equations; title: Solving the Shallow Water Equations; title: Example Scheme to Solve the SWE; title: Exercise 7; title: Appendix A: Definition and Properties of Order; title: Definition of O(h); title: The Meaning of O(h); title: Properties of O(h); title: Explanation of the Properties of O(h); title: Exercise A; title: Appendix B: Boundary Conditions; title: Introduction; title: Boundary Conditions; title: Specifying Ghost and Boundary Values; title: Common Boundary Conditions; title: Exercise B; title: Appendix C: Consistency, Convergence and Stability; title: Introduction; title: Convergence; title: Consistency and Scheme Order; title: Stability; title: Exercise C; title: Appendix D: Convergence Analysis for Iterative Methods; title: Introduction; title: Jacobi Iteration; title: Gauss-Seidel Iteration; title: SoR Iterative Scheme; title: Theory for Dominant Eigenvalues; title: Rates of Convergence of Iterative Schemes; title: Exercise D