The course starts with an introduction to Python, a computer language widely used in scientific research. Next, the three main groups of mathematical problems are dealt in three separate projects. The setup of the course is very “hands-on”: in the introductory lecture for each project the main concepts are explained, the remainder of the time is reserved for computer practica. For each project, you will write your own numerical program, analyze and visualise the results and write a brief report.
Project 1: In the first project, we will use basic numerical methods like integration, differentiation, and finding roots to solve simple physics problems. We will use a root-finding method to determine the angle with which an object should be launched in space in order to reach a certain position. Subsequently, we will use numerical integration of an ordinary differential equation to determine the motion (or phase space trajectory) of a chaotic pendulum and of a classical multi-particle system. We will compare the time averages of an observable with ensemble averages by using Monte Carlo simulations on the same system.
Project 2: Numerical integration of ordinary differential equations (ODEs)
The motion of particles that interact with each other, for example, through gravitation, can be described with ODE’s if relativistic effects are neglected. In this second project, the orbits in a three-body system is analysed and, time permitting, next the structure of stellar interiors. Each case illustrates one of the two standard ways of implementing the numerical integration of ODEs, initial value problem and boundary value problem. In the first problem, we use the restricted three-body system model to determine the orbits of the astronomical objects using initial conditions, location of objects at initial time. In the second problem, we use the polytrope equation, an approximation to the equation of state for compact stellar objects, such as white dwarfs and neutron stars, to determine the pressure and mass density in their interior as a function of the distance to their center using boundary conditions, on their surface or center.
Project 3: Numerical integration of partial differential equations (PDEs)
PDEs describe the evolution of every continuous quantity, e.g., fluid flow, magnetic fields, probability density functions. In the last project, PDEs are numerically solved by discretization of space and time. We discuss:
· The difference between implicit and explicit time discretization methods.
· The accuracy of the applied time and space discretisation.
· An analytical method to assess the stability of a discretisation.
· The impact of time step length, grid resolution and applied discretisation on the quality of the model results.
This theory will be applied on two simple problems and one more complex problem, for example, the shallow water equations.