Introductory Finite Difference Methods for PDEs

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 http://www.scilab.org/) 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:

http://www2.docm.mmu.ac.uk/STAFF/C.Mingham/

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

http://www2.docm.mmu.ac.uk/STAFF/C.Mingham/

Preface

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