### 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 1-3 are based on  and our programming course slides, 4-6 are basically the first few lectures of .

### 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, 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. , i.e., Rannacher, 2017, chapter 7 - in german

Lit. , 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.

Spring School on the Introduction on Numerical Modelling of Differential Equations - UNALM 2019