Spring School on the Introduction to Numerical Modelling of Differential Equations
Precourse October 23  25, 2019
Ole Klein (HD), Jan Philipp Thiele (LUH), Robin Görmer (LUH)
Language : Englisch

Class 1: Scientific modelling and scientific computing (high level overview), best practices for software development in scientific computing.

Class 2: Introduction to C++: functions, variables, arithmetic, recursion, loops.

Class 3: Objects, classes and containers, introduction to git.

Class 4: Floating point numbers, differential condition analysis, (forward) rounding error analysis, stability.

Class 5: Truncation error, cancellation, error estimates and asymptotics, difference quotients and swinging pendulum as simple ODE model.

Class 6: Pipe network / electric circuit to motivate large linear systems, Poisson equation as limit, extension to heterogeneous media.
Information:
Classes 13 are based on [8] and our programming course slides, 46 are basically the first few lectures of [9].
School October 28  31, 2019
Lectures: Peter Bastian (HD), Thomas Wick (LUH)
Exercises: Ole Klein, Jan Philipp Thiele, Robin Görmer
Language : Englisch

Class 1: Introduction to numerical modeling (what is scientific computing, examples of differential equations, population models, predatorprey models, Nbody 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 wellknown numerical schemes: forward Euler, backward Euler, trapezoidal rule; higherorder methods: Taylor and RungeKutta methods.

Class 4: Brief numerical analysis: discretization error, stability, convergence for initialvalue problems. In case, there is time, we deliver proofs for general RungeKutta 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, LaxMilgram, 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.