Contents

Contents Summary

Title Page
Contents
1. Introduction
2. Key Idea
3. Root finding in one dimension
4. Linear equations
5. Numerical integration
6. First order ordinary differential equations
7. Higher order ordinary differential equations
8. Partial differential equations

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
:::: :::: 1.6.3. Copyright
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
:::: 5.5. Quadratic triangulation
:::: 5.6. Romberg integration
:::: 5.7. Gauss quadrature
:::: 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
:::: :::: 8.3.6. ADI
:::: 8.4. Advection
:::: :::: 8.4.1. Upwind differencing
:::: :::: 8.4.2. Courant number
:::: :::: 8.4.3. Numerical dispersion
:::: :::: 8.4.4. Shocks
:::: :::: 8.4.5. Lax-Wendroff
:::: :::: 8.4.6. Conservative schemes

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Stuart Dalziel, last page update: 17 February 1998