The (hand-written) lecture notes can be found below. Most of the lectures roughly follow
D. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differential Equations, Springer.
Introduction of initial value problems for ordinary differential equations. Existence and uniqueness. Continuous dependence on initial data.
Euler's method: rationale, local truncation error (LTE), global error, weaknesses. (Four Matlab scripts/functions, designed for demonstrating the weaknesses of the Euler's method, are also enclosed.)
Linear multistep methods (LMMs): examples, construction and consitency.
Linear multistep methods: zero-stability and (global) convergence.
Linear multistep methods: stiff problems, absolute stability, region of absolute stability. (Two Matlab scripts for comparing the explicit and implicit Euler's methods for stiff problems are enclosed.)
Linear multistep methods: A-stability, absolute stability for systems (n>1), stiff systems.
Runge-Kutta methods: definition, examples, order conditions.
Runge-Kutta methods: absolute stability, stability function.
Runge-Kutta methods: absolute stability for systems (n>1).
Parabolic PDEs: (Spatial) discretization of a simple model problem, stiffness of the resulting initial value problem.
Parabolic PDEs: Crank-Nicolson method and its properties. Hyperbolic PDEs: conservation of energy.
Hyperbolic PDEs: discretization and conservation of "discrete energy".
Some additional reading related to the topic of the course: the lecture notes "Discretizing differential equations" by Timo Eirola and Olavi Nevanlinna.