Physics 3

Numerical Methods: Aims and Objectives

Numerical Methods 310

Aims:

To give an introduction to commonly used numerical methods.
To demonstrate the application of numerical methods to calculations encountered in lecture courses and in projects.
To enable students to use numerical techniques to tackle problems in physics that are not analytically soluble.

Course outline:

The course consists of 15 lectures, supplemented by 5 practical sessions using computers.

The numerical algorithms will be taught in a way that is as language independent as possible but a small amount of programming will be involved in order to give experience of the use of numerical methods, including the limitations and pitfalls. Since it is not intended to teach programming as part of the course, students will be provided with pre-written examples that they can use as templates and adapt to investigate the numerical methods involved; as a side effect these will illustrate good programming techniques. Some students will have encountered C programming in the Physics 2T class and all will have been given a course in Maple and should be able to use it as a programming language (which it is). Students will, therefore, normally choose to work within one of these languages. The use of Excel as a tool for numerical calculations will also be emphasised.

Matrices (2 lectures + 1 practical session) :

Inversion and the solution of linear simultaneous equations. Eigenvalues and eigenvectors. Reduction of a linear Physics calculation to one involving matrices. Application to normal mode problems from dynamics and quantum mechanics.

Interpolation and extrapolation (1 lecture) :

Polynomial interpolation. Splines. Smoothing data.

Numerical integration (2 lectures + 1 practical session) :

Simple methods for equally spaced nodes; e.g. Simpson's rule and the effect of step size. Gaussian quadrature. Examples from Physics of integrals that cannot be done analytically.

Random numbers (1 lecture) :

Generation of random numbers with normal, Poisson and other distributions. Simulation of data with a random error. Monte-Carlo methods for integration and simulations.

Root finding (1 lecture + 1 practical session) :

Bracketing and bisection methods. Secant method etc. Solution of non-linear systems of equations. Examples from Physics of equations without an algebraic solution; e.g. from quantum mechanics, solving the bound states of one, two and three dimensional square wells and, from waves, the solution of the normal modes of strings with awkward boundary conditions.

Minimising and maximising functions and data analysis (1.5 lectures) :

Brief description of common techniques used for the minimising of a function with more than one parameter. Fitting data by minimising chi-squared. The special case of linear parametrisations. Error matrix and the calculation of the errors for the fitted parameters.

Fast Fourier transform (1.5 lectures) :

The FFT. FFT of real functions, sine and cosine transforms. Data sampling, aliasing etc. The FFT in signal processing. Using Excel and Maple to apply the FFT to data.

Integration of ordinary differential equations (3 lectures + 1 practical session) :

Simple methods - e.g. Euler. Step size errors. Runge-Kutta methods. Using Runge-Kutta in Excel. Adaptive step size. Stiff equations. Using Maple for differential equations. Boundary value problems including two point boundary values. Examples of the numerical solution of differential equations from Physics and elsewhere; e.g. from dynamics, the solution of the motion of a spinning top and of a freely rotating body in three dimensions; from waves, the normal modes of a hanging rope with and without a mass at the end.

Partial differential equations (2 lectures + 1 practical session) :

Types of PDEs; elliptic, parabolic and hyperbolic with examples from Physics. Finite difference methods. Relaxation methods for boundary value problems.

Objectives:

After completion of the course the student should be able to:

explain the meaning of eigenvalues and eigenvectors;
demonstrate the reduction of a linear Physics calculation to one involving matrices;
demonstrate techniques for the solution of linear simultaneous equations;
explain and demonstrate the use of techniques for polynomial interpolation, spline fitting and soothing of data;
explain and demonstrate techniques for numerical integration;
explain and demonstrate techniques for the generation of random numbers with normal, Poisson and other distributions;
demonstrate the use of random number sets for simulation of data with a random error and for Monte-Carlo methods for integration and simulations;
describe and implement common methods for root-finding;
describe and implement common techniques used for the minimising of a function with more than one parameter;
demonstrate the fitting of data by minimising chi-squared;
explain and demonstrate the use of the FFT in signal processing;
explain the meaning of aliasing when sampling data;
discuss and implement simple methods for the solution of ODEs;
discuss elliptic, parabolic and hyperbolic types of PDEs, giving examples from Physics;
explain the use of finite difference methods and relaxation methods for boundary value problems.