Scientific Computing I

The module includes the following scientific computing topics:
- steps of the scientific computing simulation pipeline;
- classification of mathematical models (discrete/continuous, deterministic/stochastic, etc.);
- modeling with ordinary differential equations for the example of population growth;
- numerical solution of systems of ordinary differential equations;
- modeling with partial differential equations (PDE) for the example of fluid dynamics;
- numerical discretization methods for partial differential equations (finite elements, time stepping, grid generation);
- algorithms (grid traversal, data storage and access, matrix assembly) for the example of tree-structured grids;
- analysis of methods and results (adequacy and asymptotic behaviour of models; stability, consistency, accuracy, and convergence of numerical methods; sequential and parallel performance of simulation codes).

An outlook will be given on the following topics:
- implementation (architectures, parallel programming, load distribution, domain decomposition, parallel numerical methods);
- visualization for the example of fluid dynamics;
- embedding in larger simulation environments (example fluidstructure interactions);
- interactivity and computational steering.

At the end of the module, participants know the steps of the scientific computing pipeline. They are able to classify and derive simple models, to analyse crticial points and asymptotic behaviour, and to apply common discretization methods as well as explicit and implicit time stepping schemes to a given PDE model. They know the basic approaches and are able to analyse the adequacy and accuracy of numerical methods and underlying models. In addition, students understand typical grid generation, grid traversal, data storage, matrix assembly, parallelization, and visualization issues and understand examples for solution strategies and performance analysis measures.

Students should have basic knowledge in differential calculus and linear algebra.

This module comprises lectures and accompanying tutorials. The contents of the lectures will be taught by talks and presentations. Students will be encouraged to study literature and to get involved with the topics in depth. In the tutorials, concrete problems will be solved - partially in teamwork - and selected examples will be discussed.

Slides, whiteboard, exercise sheets

- A.B. Shiflet and G.W. Shiflet: Introduction to Computational Science, Princeton University Press
- Golub, Ortega: Scientific Computing: An Introduction with Parallel Computing, Academic Press, 1993
- Strang: Computational Science and Engineering, Cambridge University Press, 2007
- Tveito, Winther: Introduction to Partial Differential Equations - A Computational Approach, Springer, 1998
- Boyce, DiPrima: Elementary Differential Equations and Boundary Value Problems, Wiley, 1992 (5th edition)
- Stoer, Bulirsch: Introduction to Numerical Analysis, Springer, 1996

The examination consists of a written exam of 90 minutes in which students show that they are able to find solutions for problems arising in the field of scientific computing in a limited time. Assignments focusing on discretization methods will ensure that students are able to analyze the accuracy of a method and are able to discretize a given differential equation in space and in time. For examples of algorithms students show that they are able to analyze the performance and interprete the results. Questions test the student's knowledge of different parts of the simulation pipeline.