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 computer programming to implement some of the studied methods.

- Communicate ideas and results both orally and in writing.

- Acquire new knowledge and techniques through 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 provided in the bibliography and the compulsory material on eGela. These master classes will be complemented by problem-solving classes (GA), where students will apply the knowledge acquired in the theoretical classes to solve specific questions. During the seminar classes (S), students will give short presentations on selected topics.



Additionally, it is compulsory to implement computer programs in a programming language. These programming classes (GO) are designed to enable students to write simple programs to solve various problems using the methods presented.



A significant portion of the students' work must be done independently. The instructor will guide this work, encouraging students to engage in it regularly and to seek help when needed.

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: 20%

Attending seminar sessions and actively participating in solving problems, discussing solutions, and exploring applications 20%



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.



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.

Oinarrizko bibliografia

- 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.

Gehiago sakontzeko bibliografia

- 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.

Aldizkariak

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