Scientific computing

This repo contains exercises for three lectures and one lab course relevant to scientific computing.

Scientific computing 1:

The module introduces the steps of the scientific computing simulation pipeline on selected simulation scenarios, focusing especially on aspects of modelling and discretization:

• classification of mathematical models (discrete/continuous, deterministic/stochastic, etc.);
• discrete models (e.g. Markov chain models)
• 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, grids and adaptivity);
• limitations and errors encountered in models and discretized models
• adequacy and asymptotic behavior of models (stability, consistency, accuracy, and convergence of numerical methods)

Scientific computing 2:

• iterative solution of large sparse systems of linear equations:
  • relaxation methods
  • multigrid methods
  • steepest descent
  • conjugate gradient methods
  • preconditioning
• molecular dynamics simulations
  • particle-based modeling (n-body simulation)
  • algorithms for efficient force calculation
  • parallelization

Algorithms for scientific computing:

• Discrete Fourier Transform (DFT) and related transforms:
  • FFT: derivation and implementation
  • Fast discrete cosine/sine transforms: derivation and implementation via FFT
  • Applications: multi-dimensional data (images, video, audio) and FFT-based solvers for linear systems of equations
• Hierarchical numerical methods:
  • Hierarchical bases for one-and multi-dimensional problems
  • Computational cost versus accuracy; Sparse Grids
  • Applications: numerical quadrature, differential equations
  • Outlook: multigrid methods, Wavelets
• Space-filling curves:
  • Peano-and Hilbert curves: representation by algebraic and grammatical means
  • Tree-structured grids (quadtrees, octrees) and relation to space-filling curves
  • Applications: organisation of multi-dimensional data; parallel algorithms and cache oblivious algorithms

Scientific computing lab

The lab course gives an application oriented introduction to the following topics:

• discrete systems based on simple state transitions (Markov chains)
• explicit and implicit time stepping methods for ordinary differential equations
• numerical methods for stationary and instationary partial differential equations
• solvers for large, sparse systems of linear equations
• adaptivity and adaptively refined discretization grids
• applications from fluid dynamics and heat transfer.