This is a repository for the course on "Topics in high order accurate time integration methods" held in the Doctoral School at SISSA Trieste (Course Page 2024).

The code developed in the course is available in this repository under the MIT License, without any warranty.

The course aims to make the student aware of the cutting-edge algorithms used to numerically perform time integration and their properties in order to optimally choose the proper time integrator according to the considered phyisical model.

• Basics of numerical analysis
• The course is based on Python Notebooks. They can be run locally on your laptop or online through Google Colab (suggested, you just need a Google account). Choose one of the following possibilities:
  • Google Colab installed on Google Drive (all online on Google servers) (suggested for unexperienced users) is an app that runs on Google Drive of your Google account. The installation is straightforward at the opening of the first Google Colab project. Try opening the Google Colab link of Chapter 0 and run all the cells, follow the instructions that are inside the notebook. More info on Google Colab at Google Colab Main Page and FAQ of Google Colab.
  • Jupyter Notebook on your laptop (unrecommended if you do not use already python on your laptop and you are not a bit experienced with jupyter notebooks): you need to install python3 on your laptop, jupyter notebook and few modules (numpy, scipy, nodepy and matplotlib), best if installed with pip.
    • python3 : Installation with Anaconda
    • jupyter notebook: Installation with Anaconda and with pip
    • Add the modules: numpy, scipy, nodepy and matplotlib: with pip pip install package-name with conda conda install package-name.
    • Run the notebook Guide
    • Clone or download this repository (top right green button)
    • Test the installation with the Chapter 0 Notebook

The couse can be attended online at this link.

• Tuesday, March 12, 11:00 to 13:00
• Tuesday, March 12, 14:00 to 16:00
• Wednesday, March 13, 11:00 to 13:00
• Wednesday, March 13, 14:00 to 16:00
• Thursday, March 14, 14:00 to 18:00

• Chapter 0: Test of the installation Notebook or Google Colab
• Chapter 1: Theory of ODEs: examples, existence and uniqueness of solutions. Notebook or Google Colab
• Chapter 2: Explicit and implicit Euler, convergence, stability analysis, properties. Notebook or Google Colab
• Chapter 3: High order classical methods Notebook or Google Colab
  • Runge--Kutta methods: construction, explicit, implicit, error and stability analysis, properties, the Butcher tableau.
  • Multistep methods: explicit, implicit methods, error and stability analysis, convergence.
• Chapter 4: Entropy/energy conservative/dissipative high order schemes: relaxation Runge--Kutta methods. Notebook or Google Colab
• Chapter 5: Iterative explicit arbitrarily high order methods. Deferred Correction (DeC), Arbitrary Derivative (ADER) methods, properties, stability, convergence analysis, similarities. Notebook or Google Colab and Slides on ADER and DeC
• Chapter 6: Unconditionally positivity preserving schemes and strong stability preserving schemes. Modified Patankar methods and strong strability preserving Runge Kutta and multistep methods. Notebook or Google Colab and Slides for modified Patankar

They are in folder solutions You can open them in Google Colab through

1. Google Colab solutions
2. Google Colab solutions
3. Google Colab solutions
4. Google Colab solutions
5. Google Colab solutions
6. Google Colab solutions

Everything is provided as-is and without warranty. Use at your own risk!