Course Template
Simulating Dynamic Systems with ODEs (Euler → RK4 → Ecosystem → Stability)
Numerical Simulation of ODEs using Euler/RK4 and Analysis of Stability.
Educational Goals
This course template systematically introduces numerical simulation of dynamic systems based on ordinary differential equations. The focus is on building competencies in numerical integration, modeling dynamic processes, and analyzing stability properties. Learners connect mathematical models with their algorithmic implementation and systematically investigate the effects of step size, methods, and parameters. The template is anchored in the field of differential equations, numerical methods, and dynamic systems.
Competency Focus
- Numerical Solution of Ordinary Differential Equations using Euler and Runge-Kutta Methods
- Analysis of the Influence of Step Sizes on Accuracy and Stability of Numerical Methods
- Modeling physical and biological systems through differential equations
- Comparison of different numerical integration methods using error and difference analysis
- Visualization of dynamic processes in time series and phase portraits
- Statistical evaluation of simulation results (mean, standard deviation, range)
- Investigation of stability properties using the Jacobian matrix and eigenvalue analysis
Structure of building blocks
Coffee Cooling: Euler-ODE and dt comparison
Introduction to the numerical solution of an ODE using the Euler method, as well as analysis of the influence of the time step on simulation results.
Falling Objects: Euler vs RK4 and dt sensitivity
Comparison of Euler and RK4 methods using a physical model, as well as investigation of accuracy and sensitivity to the time step.
Lotka–Volterra with RK4 and Statistics
Simulation of a coupled ODE system using RK4, as well as evaluation of dynamics through statistical indicators and time series analysis.
Lotka-Volterra: Phase Portrait with Fixed Point Marking
Extension for analysis in the state space through phase portraits and identification of fixed points in a nonlinear system.
Stability at the Fixed Point: Jacobi & Eigenvalues
Analysis of local stability through linearization using the Jacobi matrix and interpretation of eigenvalues in the context of dynamic systems.
The clearly structured modules enable step-by-step competence development and are compatible with further topics in numerics and modeling.
Test the template in class and adapt the individual modules specifically to your learning group.