NS-256B7.5 ECTSQ2DutchBachelor
Numerical methods
FaculteitFaculty of Science
NiveauBachelor
Studiejaar2026-2027
Beschrijving
Course goals
• Knows the relevance of numerical methods for physics and astronomy.
• Can use basic numerical methods for integration, differentiation and finding roots.
• Can integrate numerically systems of first order ordinary differential equations for both initial value problems and boundary value problems.
• Can numerically solve partial differential equations of intermediate complexity using elementary techniques.
• Can assess the effect of the numerical method applied on numerical stability and on the validity of the results.
• Is able to analyse and report results of numerical experiments.
• Will have had direct experience in programming in Python.
Content
The course is divided into 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 generate Newton’s fractal. Subsequently, we will use numerical integration of an ordinary differential equation to determine the motion (or phase space trajectory) of an inverted pendulum using feedback control or to study the firing behavior of neuron using the Hodgin-Huxley model. Finally, we will study a Hopfield model for associative memory or use an artificial neural network for regression.
Project 2: In the second project, we will focus on solving different physics systems that can be modelled using ordinary differential equations. These are either initial value problems or boundary value problems, which can be numerically solved using (a combination of) the methods introduced in the first part of this course. The focus in this project is to leverage your code not just to find a numerical solution, but also to efficiently extract physical insights and analyse these systems.
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.
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 generate Newton’s fractal. Subsequently, we will use numerical integration of an ordinary differential equation to determine the motion (or phase space trajectory) of an inverted pendulum using feedback control or to study the firing behavior of neuron using the Hodgin-Huxley model. Finally, we will study a Hopfield model for associative memory or use an artificial neural network for regression.
Project 2: In the second project, we will focus on solving different physics systems that can be modelled using ordinary differential equations. These are either initial value problems or boundary value problems, which can be numerically solved using (a combination of) the methods introduced in the first part of this course. The focus in this project is to leverage your code not just to find a numerical solution, but also to efficiently extract physical insights and analyse these systems.
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.
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