Introductory Finite Difference Methods for PDEs
Publisher: Bookboon.com
Published: 2013
ISBN-13: 9788776816421
Table of contents: title: Content; title: 1. Introduction; title: 1.1. Partial Differential Equations; title: 1.2. Solution to a Partial Differential Equation; title: 1.3. PDE Models; title: 1.4. Classification of PDEs; title: 1.5. Discrete Notation; title: 1.6. Checking Results; title: 1.7. Exercise 1; title: 2. Fundamentals; title: 2.1. Taylor’s Theorem; title: 2.2. Taylor’s Theorem Applied to the Finite Difference Method (FDM); title: 2.3. Simple Finite Difference Approximation to a Derivative; title: 2.4. Example: Simple Finite Difference Approximations to a Derivative; title: 2.5. Constructing a Finite Difference Toolkit; title: 2.6. Simple Example of a Finite Difference Scheme; title: 2.7. Pen and Paper Calculation (very important); title: 2.8. Exercise 2a; title: 2.9. Exercise 2b; title: 3. Elliptic Equations; title: 3.1. Introduction; title: 3.2. Finite Difference Method for Laplace’s Equation; title: 3.3. Setting up the Equations; title: 3.4. Grid Convergence; title: 3.5. Direct Solution Method; title: 3.6. Exercise 3a; title: 3.7. Iterative Solution Methods; title: 3.8. Jacobi Iteration; title: 3.9. Gauss-Seidel Iteration; title: 3.10. Exercise 3b; title: 3.11. Successive Over Relaxation (SoR) Method; title: 3.12. Line SoR; title: 3.13. Exercise 3c; title: 4. Hyperbolic Equations; title: 4.1. Introduction; title: 4.2. 1D Linear Advection Equation; title: 4.3. Results for the Simple Linear Advection Scheme; title: 4.4. Scheme Design; title: 4.5. Multi-Level Scheme Design; title: 4.6. Exercise 4a; title: 4.7. Implicit Schemes; title: 4.8. Exercise 4b; title: 5. Parabolic Equations: the Advection-Diffusion Equation; title: 5.1. Introduction; title: 5.2. Pure Diffusion; title: 5.3. Advection-Diffusion Equation; title: 5.4. Exercise 5b; title: 6. Extension to Multi-dimensions and Operator Splitting; title: 6.1. Introduction; title: 6.2. 2D Scheme Design (unsplit); title: 6.3. Operator Splitting (Approximate Factorisation); title: 7. Systems of Equations; title: 7.1. Introduction; title: 7.2. The Shallow Water Equations; title: 7.3. Solving the Shallow Water Equations; title: 7.4. Example Scheme to Solve the SWE; title: 7.5. Exercise 7; title: 8. Appendix A: Definition and Properties of Order; title: 8.1. Definition of O(h); title: 8.2. The Meaning of O(h); title: 8.3. Properties of O(h); title: 8.4. Explanation of the Properties of O(h); title: 8.5. Exercise A; title: 9. Appendix B: Boundary Conditions; title: 9.1. Introduction; title: 9.2. Boundary Conditions; title: 9.3. Specifying Ghost and Boundary Values; title: 9.4. Common Boundary Conditions; title: 9.5. Exercise B; title: 10. Appendix C: Consistency, Convergence and Stability; title: 10.1. Introduction; title: 10.2. Convergence; title: 10.3. Consistency and Scheme Order; title: 10.4. Stability; title: 10.5. Exercise C; title: 11. Appendix D: Convergence Analysis for Iterative Methods; title: 11.1. Introduction; title: 11.2. Jacobi Iteration; title: 11.3. Gauss-Seidel Iteration; title: 11.4. SoR Iterative Scheme; title: 11.5. Theory for Dominant Eigenvalues; title: 11.6. Rates of Convergence of Iterative Schemes; title: 11.7. Exercise D