Skip navigation Download free eBooks and textbooks

Choose a category

Introductory Finite Difference Methods for PDEs

Introductory Finite Difference Methods for PDEs
3.2 (17 reviews) Read reviews
ISBN: 978-87-7681-642-1
1 edition
Pages : 144
Price: Free

Download for FREE in 4 easy steps...

We are terribly sorry, but in order to download our books or watch our videos, you will need a browser that allows JavaScript.
After entering your email address, a confirmation email will be sent to your inbox. Please approve this email to receive our weekly eBook update. We will not share your personal information with any third party.

You can also read this in Premium


This book presents finite difference methods for solving partial differential equations (PDEs) and also general concepts like stability, boundary conditions etc.

300+ Business books exclusively in our Premium eReader

  • No adverts
  • Advanced features
  • Personal library
More about Premium

Buy this eBook

Buy now

Subscribe to all 800+ eBooks

Start free trial 30 day FREE trial

About the book

  1. Description
  2. Preface
  3. About the Author
  4. Content
  5. Embed


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. Each chapter has written and computer exercises with web links to worked solutions, programs, A/V presentations and case studies. Emphasis is on the practical and students are encouraged to do numerical experiments. This book is intended for undergraduates who know Calculus and introductory programming.


The following chapters contain core material supported by pen and paper exercises together with computer-based exercises where appropriate. In addition there are web links to:

- worked solutions,
- computer codes,
- audio-visual presentations,
- case studies,
- further reading.

Codes are written using Scilab (a Matlab clone, downloadable for free from and also Matlab.

The emphasis of this book is on the practical: students are encouraged to experiment with different input parameters and investigate outputs in the computer-based exercises. Theory is reduced to a necessary minimum and provided in appendices. Web links are found on the following web page:

This book is intended for final year undergraduates who have knowledge of Calculus and introductory level computer programming.



1. Introduction
1.1 Partial Differential Equations
1.2 Solution to a Partial Differential Equation
1.3 PDE Models
1.4 Classification of PDEs
1.5 Discrete Notation
1.6 Checking Results
1.7 Exercise 1

2. Fundamentals
2.1 Taylor’s Theorem
2.2 Taylor’s Theorem Applied to the Finite Difference Method (FDM)
2.3 Simple Finite Difference Approximation to a Derivative
2.4 Example: Simple Finite Difference Approximations to a Derivative
2.5 Constructing a Finite Difference Toolkit
2.6 Simple Example of a Finite Difference Scheme
2.7 Pen and Paper Calculation (very important)
2.8 Exercise 2a
2.9 Exercise 2b

3. Elliptic Equations
3.1 Introduction
3.2 Finite Difference Method for Laplace’s Equation
3.3 Setting up the Equations
3.4 Grid Convergence
3.5 Direct Solution Method
3.6 Exercise 3a
3.7 Iterative Solution Methods
3.8 Jacobi Iteration
3.9 Gauss-Seidel Iteration
3.10 Exercise 3b
3.11 Successive Over Relaxation (SoR) Method
3.12 Line SoR
3.13 Exercise 3c

4. Hyperbolic Equations
4.1 Introduction
4.2 1D Linear Advection Equation
4.3 Results for the Simple Linear Advection Scheme
4.4 Scheme Design
4.5 Multi-Level Scheme Design
4.6 Exercise 4a
4.7 Implicit Schemes
4.8 Exercise 4b

5. Parabolic Equations: the Advection-Diffusion Equation
5.1 Introduction
5.2 Pure Diffusion
5.3 Advection-Diffusion Equation
5.4 Exercise 5b

6. Extension to Multi-dimensions and Operator Splitting
6.1 Introduction
6.2 2D Scheme Design (unsplit)
6.3 Operator Splitting (Approximate Factorisation)

7. Systems of Equations
7.1 Introduction
7.2 The Shallow Water Equations
7.3 Solving the Shallow Water Equations
7.4 Example Scheme to Solve the SWE
7.5 Exercise 7

Appendix A: Definition and Properties of Order
A.1 Definition of O(h)
A.2 The Meaning of O(h)
A.3 Properties of O(h)
A.4 Explanation of the Properties of O(h)
A.5 Exercise A

Appendix B: Boundary Conditions
B.1 Introduction
B.2 Boundary Conditions
B.3 Specifying Ghost and Boundary Values
B.4 Common Boundary Conditions
B.5 Exercise B

Appendix C: Consistency, Convergence and Stability
C.1 Introduction
C.2 Convergence
C.3 Consistency and Scheme Order
C.4 Stability
C.5 Exercise C

Appendix D: Convergence Analysis for Iterative Methods
D.1 Introduction
D.2 Jacobi Iteration
D.3 Gauss-Seidel Iteration
D.4 SoR Iterative Scheme
D.5 Theory for Dominant Eigenvalues
D.6 Rates of Convergence of Iterative Schemes
D.7 Exercise D


Choose color
Implementation code. Copy into your own page
Embed Frame - Terms of Use

The embed frame is free to use for private persons, universities and schools. It is not allowed to be used by any company for commercial purposes unless it is for media coverage. You may not modify, build upon, or block any portion or functionality of the embed frame, including but not limited to links back to the website.

The Embed frame may not be used as part of a commercial business offering. The embed frame is intended for private people who want to share eBooks on their website or blog, professors or teaching professionals who want to make an eBook available directly on their page, and media, journalists or bloggers who wants to discuss a given eBook

If you are in doubt about whether you can implement the embed frame, you are welcome to contact Thomas Buus Madsen on and seek permission.