The main focus of the module is on learning working methods. 1-3 problems of interdisciplinary relevance are selected and scientific theory, algorithmics, numerics and efficiency are rigorously practiced on these examples. In the computer exercises, students work in teams to develop, test and optimize implementations of the problems. Examples of suitable problems are e.g.:

• Wave phenomena and spectral analysis methods: Waves and oscillations in physics, the Fourier and Laplace transforms, discretization, DFT, FFT, implementation, stability analysis, duration analysis, code optimization, hardware acceleration

• Shallow water equations, asymptotic analysis, and numerical methods for geophysical flows: Nonlinear waves in fluid dynamics, characteristic theory, balanced flows and quasi-geostrophic approximation, upwind techniques for nonlinear waves and advection, semi-implicit time integration for balanced solutions.

• Gravitation, electrostatics and computational procedures: gravitation problems and Coulomb‘s law, periodic systems and convergence, Ewald summation, error analysis, Particle Mesh Ewald, efficient implementation, hardware acceleration

• Thermal conductivity equation, Poisson’s equation and solution methods: thermal conductivity equation, Poisson’s equation, parabolic PDEs, PDEs, analytical solutions for special cases, domain decomposition / finite element approximation, solution using algebraic methods, implementation, convergence analysis, code optimization, hardware acceleration

• Data analysis and dimensional reduction: examples of correlated high-dimensional signals, Rayleigh quotient and optimality principle, eigenvalue problem, singular value decomposition and usual solution methods, Nyström approximation and sparse sampling, efficient implementation