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• Class 1: Introduction to numerical modeling (what is scientific computing, examples of differential equations, population models, predator-prey models, N-body problem (energy conservation, Hamiltonian Systems), chemical networks, electrical networks, ...)

 

         Numerical methods for ODEs

 

• Class 2: A brief classification of differential equations (first order, second order, linear/nonlinear, differential algebraic systems).

          Model problem u' = f(t,u)

 

• Class 3: Derivation and analysis of three well-known numerical schemes: forward Euler, backward Euler, trapezoidal rule; higher-order methods: Taylor and Runge-Kutta methods.

 

• Class 4: Brief numerical analysis: discretization error, stability, convergence for initial-value problems. In case, there is time, we deliver proofs for general Runge-Kutta methods.

         Practical demonstrations: Numerical simulations and discussions to discretization errors, stability, and convergence. For instance using a simple                      realization of a population model.

 

• Class 5: Galerkin methods for ODEs (i.e., methods for stiff ODEs), time step adaptivity,

         Lit. [3], i.e., Rannacher, 2017, chapter 7 - in german

         Lit. [1], chapter 10.6: Galerkin; chapter 10.7: error analysis and time step control)

 

        Numerical methods for PDEs

 

• Class 6: Introduction to partial differential equations (PDEs), Modelling with partial differential equations: conservation laws, type classification of 2nd order linear PDEs.

 

• Class 7: Variational formulations, Lax-Milgram, basics in functional analysis.

 

• Class 8: Conforming finite element method, Pk finite element space, Lagrange basis, linear system, basic error analysis, interpolation error.

 

• Class 9: Finite elements on a practical level, assembling the linear system, iterative solution of linear systems.