These are implementations for the compact course "Introduction to parallel-in-time and other new time-stepping methods" at TUM. These implementations may be incomplete or false.

You need python3 and some libraries to run it. Anaconda should also work. To install the libraries run

pip3 install numpy
pip3 install scipy
pip3 install matplotlib

The implementation in RungeKutta.py is an implicit implementation where the Butcher table is created by Gauss-Lobatto. To then calculate the Lagrange polynomials, symbolic library "sympy" is used.

The relation between order and stages, using Gauss-Lobatto, is s = o+2 /2, where s is the stages and o the order.

The remaining error using Runge-Kutta is dependent on the order and of quadratic form. In log log space, the quadratic form is linear. For example, a second order remaining term has a slope of 2. A plot of different order can be seen below.

Parareal is an algorithm for parallelizing time integration. Two grids are used, a coarse one and a finer one, with two different integration schemes. In my implementations I use the Runge-Kutta with different orders. More information about parareal can be found here and here.

With higher iterations, K, the error gets reduced.