Modul

The module will be completed by an oral exam (about 20 min).

After attending the course, students understand the central properties of finite-dimensional Hamilton systems (energy conservation, symplectic flow, first integrals etc.). They know important classes of geometric time integrators such as, e.g., symplectic (partitioned) Runge-Kutta methods, splitting methods, SHAKE and RATTLE. They are not only able to implement these methods and apply them to practice-oriented problems, but also to analyze and explain the observed long-time behavior (e.g. approximative energy conservation over long times).

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• Newtonian equation of motion, Lagrange equations, Hamilton systems
• Properties of Hamilton systems: symplectic flow, energy conservation, other conserved quantities
• Symplectic numerical methods: symplectic Euler method, Störmer-Verlet method, symplectic (partitioned) Runge-Kutta methods
• Construction of symplectic methods, for example by composition and splitting
• Backward error analysis and energy conservation over long time intervals
• Mechanical systems with constraints

Familiarity with ordinary differential equations and Runge-Kutta methods (construction, order, stability, etc.) are strongly recommended. The course "Numerical methods for differential equations" provides an excellent basis. Moreover, programming skills in MATLAB are strongly recommended.

Total workload: 180 hours

Attendance: 60 hours

• lectures, problem classes, and examination
  

Self-studies: 120 hours

• follow-up and deepening of the course content,
• work on problem sheets,
• literature study and internet research relating to the course content,
• preparation for the module examination