The Discontinuous Galerkin Method for CFD Applications: Theory and Implementation (DG-CFD)

The theoretical part of the course covers the basic ideas in fluid mechanics (i.e. the compressible Euler and Navier-Stokes equations) and their numerical discretization using the discontinuous Galerkin finite element method.
The practical part of the course deals with computer implementation in a compiled language (Fortran).

After successful completion of the module, students are able to:
* understand the governing equations of fluid dynamics
* understand the basic principles of the discontinuous Galerkin (DG) method
* apply the DG method to computational fluid dynamics (CFD) test problems
* create a working computer implementation of the DG method for CFD applications

Students should have a basic knowledge of numerical mathematics (interpolation, numerical integration, approximation of ordinary differential equations) as well as some knowledge concerning computer programming languages (any language among Matlab/Octave, Python, R, C, C++, Java, Fortran, Cobol, Pascal, etc.).
Knowledge of the finite element method and of continuous mechanics are useful but not required, as well as familiarity with Unix systems (Linux, Mac).

The module includes frontal lectures at the blackboard (1 hour/week) and practicals at the computer (2 hours/week). The frontal lectures will cover the theoretical aspects of the module. During the practicals, the numerical implementation of the discontinuous Galerkin method in considered, including aspects such as makefiles, compiler flags, unit testing and version control. During the practicals, some example programs are provided and discussed; students will be then required to work on these programs to complete or extend them. Each practical is focused on a specific aspect of the implementation of the
DG method: time discretization, spatial grid, numerical integration, basis functions, differential operators and so on, so that by the end of the module the students should be able to control every aspect of a simulation code. In this way, students can deepen their understanding of the methods and concepts taught in the lectures, and gather the expertise required to work at the project for the final examination. Work in small groups, both during the practicals and for the final project, is welcome but not mandatory. Questions and discussions are strongly encouraged during both the frontal lectures and the practicals.

Blackboard, computers

Theory and Practice of Finite Elements, Alexandre Ern, Jean-Luc Guermond.
Mathematical Aspects of Discontinuous Galerkin Methods, Daniele Antonio Di Pietro, Alexandre Ern.
Implementing Spectral Methods for Partial Differential Equations, David Kopriva.
Modern Fortran Explained, Michael Metcalf, John Reid, and Malcolm Cohen.

Candidates are required to work on a small project where the concepts discussed during both lectures and practicals are applied to the numerical simulation of a CFD test problem. The final evaluation consists in the oral presentation and discussion of the project (50-60 minutes), including: a) discussion at the blackboard of the theoretical aspects of the project; b) discussion of the implementation; and c) discussion of the numerical results. Questions on topics not directly related to the project might also be part of the examination.
Candidates should demonstrate: a) knowledge of the theoretical aspects of the discontinuous Galerking method for computational fluid dynamics applications (33%); b) capability of translating the theoretical knowledge into working code (33%); and c) a critical viewpoint in analyzing the numerical results (33%).
Projects can be developed either by a single candidate or by a small group (max three people) - regardless of the group composition, the final examination will be individual.