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2019/2020

Numerical integration of partial differential equations

Code: 100121 ECTS Credits: 6
Degree Type Year Semester
2500149 Mathematics OT 4 0

Contact

Name:
Angel Calsina Ballesta
Email:
Angel.Calsina@uab.cat

Use of Languages

Principal working language:
catalan (cat)
Some groups entirely in English:
No
Some groups entirely in Catalan:
No
Some groups entirely in Spanish:
Yes

Teachers

Joan Carles Artés Ferragud

Prerequisites

This ourse has no theoretical prerequisites, although having studied partial differential equations  and / or numerical analysis will help to give context. For the practical part, there is a need  of slight familiarity with the use of programming language C for scientific computing.

Objectives and Contextualisation

 

 

Partial differential equations (PDEs) are present in most mathematical models of physical processes. As with ordinary differential equations, closed formulas are available for their solution in very few cases. That is why, in almost all applications, numerical methods are required to approximate the solutions.

This course is an introduction to the numerical methods for the approximation of the solution of PDEs. It will focus on the development and analysis of finite difference methods and finite elements for "classical" equations (transport, waves, heat and potential)

Competences

  • Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
  • Calculate and reproduce certain mathematical routines and processes with agility.
  • Develop critical thinking and reasoning and know how to communicate it effectively, both in one’s own languages and in a third language.
  • Formulate hypotheses and devise strategies to confirm or reject them.
  • Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  • Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.

Learning Outcomes

  1. Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
  2. Develop critical thinking and reasoning and know how to communicate it effectively, both in one’s own languages and in a third language.
  3. Devise demonstrations of mathematical results of numeric calculus and numeric integration of PDEs.
  4. Know how to numerically integrate ordinary differential equations and partial derivative equations.
  5. Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  6. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.

Content

 

1- Hyperbolic evolution problems. Finite differences schemes for the transport equation and conservation laws. The concepts of consistency, stability and convergence. The condition of Courant-Friedrichs-Lewy.

  2-Parabolic evolution problems. Explicit and implicit finite differences schemes. Stability. The scheme of Crank-Nicolson.

  3-Elliptical problems. Poisson problem. Stationary problems. Variational formulation. The method of Galerkin. Finite element method. Triangulations
.

 

Methodology

The classes of theory and problems will be carried out in a normal classroom of the faculty. In them the presentation of theoretical aspects of the numerical methods and their basic properties with the resolution of problems of a theoretical nature will be undertaken. Students will work on lists of problems that will be provided throughout the course.

The practical classes will be carried out in a computer classroom of the faculty. During these sessions, students will solve some applied type problem through the implementation in a programming language of some of the methods studied in the subject. These practical sessions will be evaluated from the delivery at the end of the course (the date will be announced) of the code and a practical report.

Activities

Title Hours ECTS Learning Outcomes
Type: Directed      
Exercise classes 10 0.4 1, 2, 3, 6, 5, 4
Practical classes 14 0.56 1, 2, 3, 6, 5, 4
Theory classes 26 1.04 1, 2, 3, 6, 4
Type: Autonomous      
Problems solving and practices 44 1.76 1, 2, 3, 6, 5, 4
Study 50 2 1, 2, 3, 6, 4

Assessment

The following evaluation activities will be carried out:

    Partial exam (EP). Exam with theoretical questions and problems similar to those worked during the course.
    Final Exam (EF). Examination of the whole subject with theoretical questions and problems similar to those worked during the course. It is a requirement to pass the subject that the grade of the final exam is equal to or greater than 3.5.
    Note of Practices (Prac). It will be evaluated based on the project (program) and the practice report. It is a requirement to pass the subject that the qualification of the practices is equal to or greater than 3,5.


The final grade will be obtained using the formula

         QF = (25EP + 40EF + 35Prac) / 100;

Additionally, students will be able (and will be recommended) to deliver some problems of the list of problems that will consist of experimenting with computer on the properties of some of the numerical methods that will be seen during the course. Evaluation of these problems may add a point (out of 10) to the EF rating.

There will be a resitting exam with the same format as the EF exam. The practices are not recoverable.

The "matrícula de honor" will be awarded after the first evaluation such that the subject can be passed.

Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Final exam 0.4 3 0.12 1, 2, 3, 5, 4
Partial exam 0.25 3 0.12 1, 2, 3, 5, 4
Practice delivery 0.35 0 0 1, 2, 6, 5, 4

Bibliography

Bibliography

  • J. C. Strikwerda: Finite difference schemes and partial differential equations, SIAM, 2004
  • K.W. Morton, D.F. Mayers: Numerical Solution of Partial Differential Equations, Cambridge University Press, 1994.
  • M. G. Larson, F. Benzgon: The finite element method: Theory, implementation and applications. Springer, 2013.
  • Josep Masdemont: Curs d’elements finits amb aplicacions. Edicions UPC, 2002.
  • D. R. Lynch: Numerical Partial Differential Equations for Environmental Scientists and Engineers, Springer, 2005

 

Aditional bibliography

  • P. G. Ciarlet: The Finite element methods for elliptic problems. North Holland, 1979.
  • C. Johnson: Numerical Solution of Partial Differential Equations by the Finite Element method, Cambridge University Press, 1994.
  • L. Lapidus, G.F Pinder:  Numerical solution of partial differential equations in science and engineering, John Wiley & Sons, 1982.
  • Leveque,R.J.: Finite difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
  • P.A. Raviart, J.M. Thomas: Introduction à l’analyse numérique des équations aux dérivées partielles, Masson, 1983.