Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OT | 4 | 2 |
You can check it through this link. To consult the language you will need to enter the CODE of the subject. Please note that this information is provisional until 30 November 2023.
This course has no theoretical prerequisites, although courses on partial differential equations and numerical analysis would help to provide context. The practical work requires a minimum familiarity with the use of the C programming language for scientific computing.
This course is an introduction to numerical methods for the solution of partial differential equations (PDE).
PDE are in the basis of most mathematical models of physical processes. As with ordinary differential equations (ODE), closed-formulae solutions are available in very few cases. Because of that, in almost all applications numerical methods are required for the approximation of their solutions. Contrary to ODE, though, there are no general numerical methods applicable to almost all PDE except for some special behaviours: the methods are specific for small families of PDE. The ideas giving rise to the methods are general, and, in this way, we can speak of families of methods, like finite difference methods or finite element methods.
The course will be focussed on the development and analysis of finite difference and finite element methods for the classical PDE (transport, waves, heat and potential), although some comments will be made on other methods (such as characteristics or spectral) and other equations.
1.- Finite differences
Hyperbolic evolution problems. The transport equation. Consistency, stability and convergence. Local truncation error and order of a method. The Courant-Friedrichs-Lewy condition.
Parabolic evolution problems. Explicit and implicit methods. The method of John Crank i Phyllis Nicolson. Stability
Stationary problems. The Poisson equation.
2.- Finite elements
Variational formulation. Stages: meshing, assembly, solution of the linear system, post-processing. Example: the 2D Poisson equation.
Triangulations. Interpolation in several variables, families of finite elements. Boundary conditions. Assembly and global formulation.
The sessions on theory and problems will be carried out in a classroom. These sessions will consist in the presentation of the methods and their properties and the solutio of problems of a theoretical nature. Problem lists with be provided during the course.
The practical sessions will be carried out in a computer room. In these sessions, students will solve an applied problem through the implementation in a programming language of some of the methods studied in the course. These sessions will be evaluated from the delivery at the end of the course (the exact date will be announced) of the code and a report.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Exercise classes | 10 | 0.4 | 1, 2, 3, 8, 6, 5 |
Practical classes | 14 | 0.56 | 1, 2, 3, 8, 6, 5 |
Theory classes | 26 | 1.04 | 1, 2, 3, 8, 5 |
Type: Autonomous | |||
Problems solving and practices | 44 | 1.76 | 1, 2, 3, 8, 6, 5 |
Study | 50 | 2 | 1, 2, 3, 8, 5 |
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Delivery of problems | 0.05 | 0 | 0 | 1, 2, 3, 9, 8, 6, 5 |
Final exam | 0.45 | 3 | 0.12 | 2, 3, 9, 8, 6, 5 |
Practice delivery | 0.5 | 0 | 0 | 1, 2, 4, 9, 8, 7, 6, 5 |
Recovery exam | 0.5 | 3 | 0.12 | 2, 8, 6, 5 |
- C. Johnson: Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover, 2009.
- M. G. Larson, F. Benzgon: The finite element method: Theory, implementation and applications. Springer, 2013.
- J. Masdemont: Curs d'elements finits i aplicacions. Edicions UPC, 2002.
- R.M.M. Mattheij, S.W. Rienstra, J.H.M. ten Thije Boonkkamp: Partial Differential Equations. Modeling, Analysis, Computation. SIAM, 2005.
- K.W. Morton, D.F. Mayers: Numerical Solution of Partial Differential Equations, Cambridge University Press, 1994.
- J. C. Strikwerda: Finite difference schemes and partial differential equations, SIAM, 2004.
- Preferably a Linux environment
- code-oriented text editor (e.g. Kate)
- GNU C compiler
- gnuplot
- image manipulation tools (e.g. imagemagick)
- GNU Octave