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

• Molecular dynamics or Monte Carlo simulation: Implementation and analysis of particle-based simulations, in simple molecular systems (e.g. water, ionic solution, polymers), treating long-range electrostatics, measuring thermodynamic (e.g. heat capacity) or transport properties (e.g. diffusion coefficient).

Classes and tutorials are organized in hybrid mode:
Classes:  Mondays, Wednesdays,  10:00h-12:00h,  Seminar Room 025/026, Arnimallee 6
Part II: (Webex-Link, starting from December 05, 2022)

Proj. Seminar:  Tuesdays, Thursdays,  10:00h-12:00h,  Seminar Room 030, Arnimallee 6.

The lecture of Nov. 7 was swapped with the tutorial of Tuesday Nov. 8, and -- as an exception -- recorded (link).

Lecturer:
Part I: Prof. Dr.-Ing. Rupert Klein  (rupert.klein@math.fu-berlin.de)

Tutors:
Tom Doerffel (tom.doerffel@fu-berlin.de)
Maaike Galama (maaike.galama@fu-berlin.de)
Atharva Kelkar (atkelkar@zedat.fu-berlin.de)

Credits:

Exam:  Wednesday, February 15, 2023, 10:00h-12:00h  (time of last lecture) at Lecture Hall (ZIB) Takustr. 7.

Post-exam: Friday, April, 2023, 10:00h-12:00; Hörsaal A, Arnimallee 22

Participation:

• active: Successful presentation of your study projects in February
• continuous: Attendance of and contributions to the project tutorials and project development, respectively.

Textbooks and References

For the first part of the course (Instructor: Prof. Rupert Klein):

Lecture Script, Part I (by Prof. Nikki Vercauteren);
Lecture Script "Mathematical modelling in Climate Research"

1. E.A. Bender. An Introduction to Mathematical Modeling. Dover, Mineola, 2000.
2. E. Buckingham. On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Phys. Rev. 4, 345–376, 1914.
3. D.R. Durran. Numerical Methods for Fluid Dynamics. Texts in Applied Mathematics, Springer, 2011.
4. E. Hairer. Geometric Numerical Integration. Lecture Notes, TU Muenchen, 2010.
5. G. Hornberger and P. Wiberg. Numerical Methods in the Hydrological Sciences. Special Publication Series 57, American Geophysical Union, 2005.
6. R. Illner, C.S. Bohun, S. McCollum, and T. van Roode. Mathematical Modelling: A Case Studies Approach. AMS, Providence, 2005.
7. A. Quarteroni, R. Sacco and F. Saleri. Numerical Mathematics. Texts in Applied Mathematics, Springer, 2007.
8. S. Socolofsky and G. Jirka Special Topics in Mixing and Transport Processes in the Environment. Lecture notes, Texas A&M University and University of Karlsruhe, 2005.
9. G. Stolz Introduction au Calcul Scientifique. Lecture notes, Ecole des Mines, 2014.
10. G. Teschl. Ordinary Differential Equations and Dynamical Systems. AMS, Providence, 2012.
11. A. H. Nayfeh, Perturbation Methods, Wiley, 1973.
12. J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Series: Applied Mathematical Sciences 114, Springer Verlag, 1996.
13. W. Schneider, Mathematische Methoden in der Strömungsmechanik, Vieweg, 1978.
14. G. Hairer, Ch. Lubich, E. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 2nd ed., 2006.

For the second part of the course (Instructor: Dr. Mohsen Sadeghi):

1. The Feynman Lectures on Physics https://www.feynmanlectures.caltech.edu
2. Frenkel, D. and Smit, B. (2002). Understanding Molecular Simulation: From Algorithms to Applications (2nd ed.). Academic Press. https://doi.org/10.1016/B978-0-12-267351-1.X5000-7 (available for download from within FU network with the Institutional Access to ScienceDirect).
3. Allen, M. P. and Tildesley, D. J. (2017). Computer Simulation of Liquids (2nd ed.). Oxford Academic. https://doi.org/10.1093/oso/9780198803195.001.0001
4. Rapaport, D. (2004). The Art of Molecular Dynamics Simulation (2nd ed.). Cambridge University Press. https://doi.org/10.1017/CBO9780511816581 (Institutional Access from FU Berlin)
5. Tuckerman, M. E. (2009). Statistical Mechanics: Theory and Molecular Simulation. Oxford University Press (full-text provided by Mark Tuckerman via ResearchGate https://www.researchgate.net/publication/265327519_Statistical_Mechanics_Theory_And_Molecular_Simulation)