Contents

Title Page
1. Introduction
:::: 1.1. Objective
:::: 1.2. Books
:::: :::: General:
:::: :::: More specialised:
:::: 1.3. Programming
:::: 1.4. Tools
:::: :::: 1.4.1. Software libraries
:::: :::: 1.4.2. Maths systems
:::: 1.5. Course Credit
:::: 1.6. Versions
:::: :::: 1.6.1. Word version
:::: :::: 1.6.2. Notation in HTML formatted notes
2. Key Idea
3. Root finding in one dimension
:::: 3.1. Why?
:::: 3.2. Bisection
:::: :::: 3.2.1. Convergence
:::: :::: 3.2.2. Criteria
:::: 3.3. Linear interpolation (regula falsi)
:::: 3.4. Newton-Raphson
:::: :::: 3.4.1. Convergence
:::: 3.5. Secant (chord)
:::: :::: 3.5.1. Convergence
:::: 3.6. Direct iteration
:::: :::: 3.6.1. Convergence
:::: 3.7. Examples
:::: :::: 3.7.1. Bisection method
:::: :::: 3.7.2. Linear interpolation
:::: :::: 3.7.3. Newton-Raphson
:::: :::: 3.7.4. Secant method
:::: :::: 3.7.5. Direct iteration
:::: :::: :::: 3.7.5.1. Addition of x
:::: :::: :::: 3.7.5.2. Multiplcation by x
:::: :::: :::: 3.7.5.3. Approximating f'(x)
:::: :::: 3.7.6. Comparison
:::: :::: 3.7.7. Fortran program
4. Linear equations
:::: 4.1. Gauss elimination
:::: 4.2. Pivoting
:::: :::: 4.2.1. Partial pivoting
:::: :::: 4.2.2. Full pivoting
:::: 4.3. LU factorisation
:::: 4.4. Banded matrices
:::: 4.5. Tridiagonal matrices
:::: 4.6. Other approaches to solving linear systems
:::: 4.7. Over determined systems
:::: 4.8. Under determined systems
5. Numerical integration
:::: 5.1. Manual method
:::: 5.2. Trapezium rule
:::: 5.3. Mid-point rule
:::: 5.4. Simpson's rule
:::: 5.6. Romberg integration
:::: 5.8. Example of numerical integration
:::: :::: 5.8.1. Program for numerical integration
6. First order ordinary differential equations
:::: 6.1. Taylor series
:::: 6.2. Finite difference
:::: 6.3. Truncation error
:::: 6.4. Euler method
:::: 6.5. Implicit methods
:::: :::: 6.5.1. Backward Euler
:::: :::: 6.5.2. Richardson extrapolation
:::: :::: 6.5.3. Crank-Nicholson
:::: 6.6. Multistep methods
:::: 6.7. Stability
:::: 6.8. Predictor-corrector methods
:::: :::: 6.8.1. Improved Euler method
:::: :::: 6.8.2. Runge-Kutta methods
7. Higher order ordinary differential equations
:::: 7.1. Initial value problems
:::: 7.2. Boundary value problems
:::: :::: 7.2.1. Shooting method
:::: :::: 7.2.2. Linear equations
:::: 7.3. Other considerations
:::: :::: 7.3.1. Truncation error
:::: :::: 7.3.2. Error and step control
8. Partial differential equations
:::: 8.1. Laplace equation
:::: :::: 8.1.1. Direct solution
:::: :::: 8.1.2. Relaxation
:::: :::: :::: 8.1.2.1. Jacobi
:::: :::: :::: 8.1.2.2. Gauss-Seidel
:::: :::: :::: 8.1.2.3. Red-Black ordering
:::: :::: :::: 8.1.2.4. Successive Over Relaxation (SOR)
:::: :::: 8.1.3. Multigrid
:::: :::: 8.1.4. The mathematics of relaxation
:::: :::: :::: 8.1.4.1. Jacobi and Gauss-Seidel for Laplace equation
:::: :::: :::: 8.1.4.2. Successive Over Relaxation for Laplace equation
:::: :::: :::: 8.1.4.3. Other equations
:::: :::: 8.1.5. FFT
:::: :::: 8.1.6. Boundary elements
:::: :::: 8.1.7. Finite elements
:::: 8.2. Poisson equation
:::: 8.3. Diffusion equation
:::: :::: 8.3.1. Semi-discretisation
:::: :::: 8.3.2. Euler method
:::: :::: 8.3.3. Stability
:::: :::: 8.3.4. Model for general initial conditions
:::: :::: 8.3.5. Crank-Nicholson