Computational techniques are a critical aspect of modern physics, science and engineering. They sit apart from theoretical and experimental physics but borrow characteristics from both. The aim is to turn a computer into a virtual laboratory for research, that allows breakthroughs and insights from what would otherwise be intractable problems by analytical methods. This unit focuses specifically on the computational techniques for solving problems in physics, engineering and science in general. It is not a course in programming though a low level of programming ability will be required to practice the techniques. Topics to be covered: o Introduction to Python and the Python scientific environment. o ODEs: Euler, Runge-Kutta and adaptive techniques, examining accuracy and stability with examples drawn from planetary science and chaotic systems. o Spectral methods: systems of linear equations, spectral analysis and analysis of normal modes. o PDEs: Initial and boundary conditions, discretisation. Relaxation and implicit schemes. Examples of Poisson, diffusion and wave equations. o Monte Carlo methods: random numbers, Monte Carlo integration, random walks, Metropolis algorithm. Examples of Ising model and phase transitions. o Convex optimisation: convex sets and functions, optimisation problems, linear and quadratic programming, duality.