An edition of Introductory Finite Difference Methods for PDEs
(2013)
Introductory Finite Difference Methods for PDEs
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This book presents finite difference methods for solving partial differential equations (PDEs) and also general concepts like stability, boundary conditions etc. Material is in order of increasing complexity (from elliptic PDEs to hyperbolic systems) with related theory included in appendices.
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Publish Date
2013
Publisher
Bookboon
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Book Details
Table of Contents
Introduction
Partial Differential Equations
Solution to a Partial Differential Equation
PDE Models
Classification of PDEs
Discrete Notation
Checking Results
Exercise 1
Fundamentals
Taylor’s Theorem
Taylor’s Theorem Applied to the Finite Difference Method (FDM)
Simple Finite Difference Approximation to a Derivative
Example: Simple Finite Difference Approximations to a Derivative
Constructing a Finite Difference Toolkit
Simple Example of a Finite Difference Scheme
Pen and Paper Calculation (very important)
Exercise 2a
Exercise 2b
Elliptic Equations
Introduction
Finite Difference Method for Laplace’s Equation
Setting up the Equations
Grid Convergence
Direct Solution Method
Exercise 3a
Iterative Solution Methods
Jacobi Iteration
Gauss-Seidel Iteration
Exercise 3b
Successive Over Relaxation (SoR) Method
Line SoR
Exercise 3c
Hyperbolic Equations
Introduction
1D Linear Advection Equation
Results for the Simple Linear Advection Scheme
Scheme Design
Multi-Level Scheme Design
Exercise 4a
Implicit Schemes
Exercise 4b
Parabolic Equations: the Advection-Diffusion Equation
Introduction
Pure Diffusion
Advection-Diffusion Equation
Exercise 5b
Extension to Multi-dimensions and Operator Splitting
Introduction
2D Scheme Design (unsplit)
Operator Splitting (Approximate Factorisation)
Systems of Equations
Introduction
The Shallow Water Equations
Solving the Shallow Water Equations
Example Scheme to Solve the SWE
Exercise 7
Appendix A: Definition and Properties of Order
Definition of O(h)
The Meaning of O(h)
Properties of O(h)
Explanation of the Properties of O(h)
Exercise A
Appendix B: Boundary Conditions
Introduction
Boundary Conditions
Specifying Ghost and Boundary Values
Common Boundary Conditions
Exercise B
Appendix C: Consistency, Convergence and Stability
Introduction
Convergence
Consistency and Scheme Order
Stability
Exercise C
Appendix D: Convergence Analysis for Iterative Methods
Introduction
Jacobi Iteration
Gauss-Seidel Iteration
SoR Iterative Scheme
Theory for Dominant Eigenvalues
Rates of Convergence of Iterative Schemes
Exercise D
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- Created April 8, 2016
- 4 revisions
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| August 22, 2020 | Edited by ISBNbot2 | normalize ISBN |
| April 8, 2016 | Edited by Alice Kirk | Edited without comment. |
| April 8, 2016 | Edited by Alice Kirk | Added new cover |
| April 8, 2016 | Created by Alice Kirk | Added new book. |