ESTA ASIGNATURA SOLO SE IMPARTIRÁ EN INGLES



DESCRIPTION

The course will show the most important numerical methods and techniques of Numerical Analysis for the approximate numerical solution of differential equations, in a systematic way and with particular attention to partial differential equations. A priori properties of these algorithms such as accuracy, stability and convergence will be studied.

Even if there is no prerequisite, this course is related with the other courses of Numerical Analysis and the ones of Differential Equations.



A level of B2 or higher is recommended to attend courses taught in English.

COMPETENCES / AIM

M10CM01- Know the most important results and demonstrations of the course.

M10CM02- Know some of the advanced techniques of numerical calculus and translate them to algorithms.

M10CM03- Understand the mathematical concepts needed to solve differential equations from a numerical point of view.





RESULTS OF LEARNING

Apply the knowledge of solving differential equations to the resolution of theoretical and practical problems.

Use of computer programming in order to apply some of the studied methods.

Communicate ideas and results in oral and written way.

Achieve new knowledge and techniques in an independent learning.

THEORETICAL CONTENTS

1. MORE ABOUT NUMERICAL METHODS FOR O.D.E.

2. NUMERICAL SOLUTION FOR EVOLUTION P.D.E. USING F.F.T.

3. FINITE DIFFERENCE METHODS FOR PARABOLIC PROBLEMS.

4. FINITE DIFFERENCE METHODS FOR HYPERBOLIC PROBLEMS.

5. FINITE ELEMENT METHOD FOR ELLIPTIC PROBLEMS.

6. SPECTRAL METHODS FOR EVOLUTIONS PROBLEMS.



PRACTICAL CONTENTS

THERE WILL BE COMPUTER PROGRAMMING FOR EACH CHAPTER.

METHODOLOGY

The theoretical background will be presented in master classes (M), following the references given in the bibliography and the compulsory material of eGela. These master classes will be complemented with classes of problems (GA) where students have to solve questions, for which they have to apply the knowledge acquired in the theoretical classes. During the seminar classes (S) the students will give a short class presenting the review of some topic. Finally, it is compulsory to implement computer programs in some programming language. These programming classes (GO) are oriented in such a way that the students should be capable of writing simple programs to solve different problems using some of the presented methods.

A big part of the students' work has to be done personally. The instructor will guide such work and will encourage the students to do it regularly, as well as motivate them to ask for help if they need any.

• Sistema de Evaluación Continua
• Sistema de Evaluación Final
• Herramientas y porcentajes de calificación:
  • Read carefully the guidelines (%): 100

GUIDELINES FOR THE CONTINUOUS ASSESSMENT SYSTEM

The grading will consider the individual work, as well as the work done with computer programming and the written exam.



Exams: 40%

Computer programming: 30%

Theoretical and practical homework: 30%



RENOUNCE TO THE CONTINUOUS ASSESSMENT SYSTEM

The student must give written notice of withdrawal of continuous assessment system in a period of 9 weeks.



Article 8.3: In any case, students will have the right to be evaluated through the final evaluation system, regardless of whether or not they have participated in the continuous evaluation system. To do this, students must submit in writing to the teaching staff responsible for the subject the waiver of continuous assessment, for which they will have a period of 9 weeks for the quarterly subjects and 18 weeks for the annual subjects, starting from the beginning of the semester or course respectively, according to the academic calendar of the center.



DECLINING TO SIT

A student who does not fulfill the necessary conditions of summing up 1.5 points in the two exams or reaching 6 points before the final exam and does not take the final exam will obtain <>.



Article 12.2: In the case of continuous assessment, if the weight of the final test is greater than 40% of the grade for the course, it will suffice not to take the final test so that the final grade for the course is not presented or not filed. Otherwise, if the weight of the final test is equal to or less than 40% of the grade for the course, the student may waive the call within a period that, at least, will be up to one month before the end date of the teaching period of the corresponding subject. This resignation must be submitted in writing to the faculty responsible for the subject.



GUIDELINES FOR THE END-OF-COURSE (FINAL) ASSESSMENT

In the case of students who have not passed the evaluation of the activities carried out throughout the course (computer practices, exercises, seminars) or have chosen the final evaluation modality, in the ordinary call they must also take a complementary test designed for the evaluation of the activities carried out throughout the course. This test can consist of an oral presentation, a demonstration before a computer or a written description of the practical knowledge covered in the activities planned throughout the course. The value of this test will be taken into account in the same proportion as in the continuous evaluation.

GUIDELINES FOR THE CONTINUOUS ASSESSMENT SYSTEM

The grading will consider the individual work, as well as the work done with computer programming and the written exam.



Exams: 40%

Computer programming: 30%

Theoretical and practical homework: 30%



RENOUNCE TO THE CONTINUOUS ASSESSMENT SYSTEM

The student must give written notice of withdrawal of continuous assessment system in a period of 9 weeks.



Article 8.3: In any case, students will have the right to be evaluated through the final evaluation system, regardless of whether or not they have participated in the continuous evaluation system. To do this, students must submit in writing to the teaching staff responsible for the subject the waiver of continuous assessment, for which they will have a period of 9 weeks for the quarterly subjects and 18 weeks for the annual subjects, starting from the beginning of the semester or course respectively, according to the academic calendar of the center.



DECLINING TO SIT

A student who does not fulfill the necessary conditions of summing up 1.5 points in the two exams or reaching 6 points before the final exam and does not take the final exam will obtain <>.



Article 12.2: In the case of continuous assessment, if the weight of the final test is greater than 40% of the grade for the course, it will suffice not to take the final test so that the final grade for the course is not presented or not filed. Otherwise, if the weight of the final test is equal to or less than 40% of the grade for the course, the student may waive the call within a period that, at least, will be up to one month before the end date of the teaching period of the corresponding subject. This resignation must be submitted in writing to the faculty responsible for the subject.



GUIDELINES FOR THE END-OF-COURSE (FINAL) ASSESSMENT

In the case of students who have not passed the evaluation of the activities carried out throughout the course (computer practices, exercises, seminars) or have chosen the final evaluation modality, in the ordinary call they must also take a complementary test designed for the evaluation of the activities carried out throughout the course. This test can consist of an oral presentation, a demonstration before a computer or a written description of the practical knowledge covered in the activities planned throughout the course. The value of this test will be taken into account in the same proportion as in the continuous evaluation.

COMPULSORY MATERIAL
Theoretical material stored in the virtual class of eGela.

Bibliografía básica

- M.S. GOCKENBACH: P.D.E. Analytical and Numerical Methods, SIAM 2003.

- J.C. STRIKWERDA: Finite Diference Schemes and PDE, Wadsworth & Brooks 1989.

- L. LAPIDUS & G.F. PINDER: Numerical Solutions of PDE in science and engineering, John Wiley and Sons, 1999.

- E.H. TWIZELL: Computational Methods for P.D.E., John Wiley and Sons, 1988.

- B. FORNBERG: A Practical Guide to Pseudospectral Methods, Cambridge University Press 1998.

- A. TVEITO & R. WINTHER: Introduction to Partial Differential Equations - A Computational Approach, Springer, 1998.

- M.T. HEATH: Scientific computing: an introductory survey, Mc Graw Hill, 2002.

- V.G. GANZHA & E.V. VOROZHTSOV: Numerical solutions for Partial Differential Equations: Problem solving using Mathematica, CRC Press, 1996.

- Uri M. ASCHER: Numerical Methods for Evolutionary D. E., SIAM 2008.

- K.W. MORTON & D.F. MAYERS: Numerical Solution of PDE, Cambridge 2005.

- J.W. THOMAS: Numerical PDE. Finite Difference Methods, Springer, 1995.

- L.N. TREFETHEN: Spectral Methods in MATLAB, SIAM 2000.

Bibliografía de profundización

- J.D. LAMBERT, Numerical methods for O.D.E.: the initial value problems, Wiley, 1991.
- S.P. NORSETT, E. HAIRER & G. WANNER, Solving ordinary differential equations i: Nonstiff problems, Springer, 1987 (1993 second edition).
- E. HAIRER & G. WANNER, Solving ordinary differential equations ii: Stiff and Differential algebraic Problems, Springer, 1991.
- W. HUNDSDORFER & J.C. VERWER: Numerical Solutions of Time-Dependent Advection-Diffusion-Reaction Equations, Springer 2007.
- C. JOHNSON: Numerical solution of P.D.E. by the F.E.M., Cambridge University Press 1987.
- W.E. SCHIESSER: The numerical method of line: integration of Partial Differential equations, Academic Press, 1991.
- W.E. SCHIESSER & G.W. GRIFFTHS: A compendium of partial differential equation models: method of lines analysis with Matlab, Cambridge University Press, 2009.
- J.S. HESTHAVEN, S. GOTTLIEB & D. GOTTLIEB: Spectral methods for time-dependent problems, Cambridge University Press, 2007.
- A.R. MITCHELL & D.F. GRIFFTHS: The Finite Difference Method in Partial Differential Equations, John Wiley and Sons, 1980.
- A. QUARTERONI & A. VALLI: Numerical Approximation of Partial Differential Equations, Springer-Verlag, 1994.
- L. DEMKOWICZ: Computing with hp-adaptive finite elements, v.1, One and two dimensional elliptic and Maxwell problems, Chapman and Hall/CRC, 2007.

Revistas

JOURNALS

Mathematical Methods in the Applied Sciences
International Journal for Numerical Methods in Engineering
International Journal for Numerical Methods in Fluids
International Journal for Numerical Methods in Biomedical Engineering

Direcciones web

Videos for Korteweg de Vries equation:
https://www.youtube.com/watch?v=i7ORX97drdg
https://www.youtube.com/watch?v=VFM48pSLwGc
CIMNE: International Center for Numerical Methods in Engineering:
http://www.cimne.upc.es/
NAG Library:
http://www.nag.co.uk/
IMSL Library:
http://www.roguewave.com/products-services/imsl-numerical-libraries
SIAM Journal of Numerical Analysis:
http://epubs.siam.org/SINUM