Guías Docentes Electrónicas
1. General information
Course:
NUMERICAL ANALYSIS OF PDE AND APPROXIMATION
Code:
310934
Type:
ELECTIVE
ECTS credits:
6
Degree:
2351 - MASTER DEGREE PROGRAMME IN PHYSICS AND MATHEMATICS-FISYMAT
Academic year:
2020-21
Center:
602 - E.T.S. INDUSTRIAL ENGINEERING OF C. REAL
Group(s):
20 
Year:
1
Duration:
First quarter
Main language:
Spanish
Second language:
English
Use of additional languages:
English Friendly:
Y
Web site:
Bilingual:
N
Lecturer: DAMIAN CASTAÑO TORRIJOS - Group(s): 20 
Building/Office
Department
Phone number
Email
Office hours
Edificio Sabatini / 1.53
MATEMÁTICAS
925268800 ext 5722
Damian.Castano@uclm.es

Lecturer: HENAR HERRERO SANZ - Group(s): 20 
Building/Office
Department
Phone number
Email
Office hours
Francisco Fdez. Iparraguirre/341
MATEMÁTICAS
926295412
henar.herrero@uclm.es

Lecturer: MARIA CRUZ NAVARRO LERIDA - Group(s): 20 
Building/Office
Department
Phone number
Email
Office hours
Francisco Fdez Iparraguirre. Despacho 326.
MATEMÁTICAS
3469
mariacruz.navarro@uclm.es

2. Pre-Requisites

Calculus, algebra, differential equations and functional analysis.

3. Justification in the curriculum, relation to other subjects and to the profession

Partial differential equations are the main tool for modeling in science and technology. Only a few of these equations have an analytical solution. For this reason, numerical resolution is essential for scientific progress. To acquire knowledge on numerical analysis is relevant in an Applied Mathematics Master of Science.


4. Degree competences achieved in this course
Course competences
Code Description
CB06 Possess and understand knowledge that provides a basis or opportunity to be original in the development and / or application of ideas, often in a research context.
CB07 Apply the achieved knowledge and ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to the area of study
CB08 Be able to integrate knowledge and face the complexity of making judgments based on information that, being incomplete or limited, includes reflections on social and ethical responsibilities linked to the application of knowledge and judgments
CB09 Know how to communicate the conclusions and their supported knowledge and ultimate reasons to specialized and non-specialized audiences in a clear and unambiguous way
CB10 Have the learning skills which allow to continue studying in a self-directed or autonomous way
CE01 Solve physical and mathematical problems, planning their solutions based on the available tools and time and resource constraints
CE02 Develop the ability to decide the appropriate techniques to solve a specific problem with special emphasis on those problems associated with the Modeling in Science and Engineering, Astrophysics, Physics, and Mathematics
CE05 Know how to obtain and interpret physical and/or mathematical data that can be applied in other branches of knowledge
CE07 Ability to understand and apply advanced knowledge of mathematics and numerical or computational methods to problems of biology, physics and astrophysics, as well as to build and develop mathematical models in science, biology and engineering
CE08 Ability to model, interpret and predict from experimental observations and numerical data
CG03 Present publicly the research results or technical reports, to communicate the conclusions to a specialized court, interested persons or organizations, and discuss with their members any aspect related to them
CG04 Know how to communicate with the academic and scientific community as a whole, with the company and with society in general about Physics and/or Mathematics and its academic, productive or social implications
CT03 Develop critical reasoning and the ability to criticize and self-criticize
CT05 Autonomous learning and responsibility (analysis, synthesis, initiative and teamwork)
5. Objectives or Learning Outcomes
Course learning outcomes
Description
Interpretation of the obtained numerical solution and critical judgment of its quality. Relation with the applied science referred to
Gain the ability to solve a specific problem as a team: from the choice of an appropriate method to the oral and written presentation of the results obtained after its implementation
Learn to use some tools of Basic Analysis and Functional Analysis to carry out the numerical analysis of a method
Know some software tools that allow to completely solve a problem in the computer, which entails know how to program, generate a computational mesh, apply the appropriate calculus module and visualize the numerical solution. Practical problem solving.
Understand the theoretic design of finite element, finite difference, finite and spectral volume methods, from known analytic techniques (variational formulations, Taylor developments, integration formulas by parts).
Understand the specific characteristics of the elliptic, parabolic and hyperbolic equations which be solved by numerical methods
Know and understand the basic concepts of consistency, stability and convergence of a numerical scheme in this context, as well as their interrelation
Additional outcomes
Not established.
6. Units / Contents
  • Unit 1: Finite Differences
    • Unit 1.1: Boundary value problems
    • Unit 1.2: Parabolic evolution problems
    • Unit 1.3: Error analysis
    • Unit 1.4: Practice exercises
  • Unit 2: Spectral Methods
    • Unit 2.1: Fourier approximation
    • Unit 2.2: Orthogonal polynomial approximation
    • Unit 2.3: Practice exercises
  • Unit 3: Finite Elements
    • Unit 3.1: Formulation and error analysis
    • Unit 3.2: Effective implementation
    • Unit 3.3: Practice exercises
  • Unit 4: Finte Volumes and Finite Differences
    • Unit 4.1: Finite Differences for hyperbolic evolution problems
    • Unit 4.2: Formulation and error analysis
    • Unit 4.3: Effective implementation
    • Unit 4.4: Practice exercises
  • Unit 5: Courses and seminars
7. Activities, Units/Modules and Methodology

All training activities will be recoverable, in other words, there must be an alternative evaluation test that allows to reassess the acquisition of the same skills in the ordinary, extraordinary and special call for completion. If exceptionally, the evaluation of any of the training activities cannot be recovered, it must be specified in the description and be expressly authorized by the department.

Training Activity Methodology Related Competences ECTS Hours As Com Description
Class Attendance (theory) [ON-SITE] Lectures 1.5 37.5 Y N
Computer room practice [ON-SITE] Problem solving and exercises 0.7 17.5 Y Y
Individual tutoring sessions [ON-SITE] 0.2 5 Y N
Writing of reports or projects [OFF-SITE] Self-study 3.6 90 Y Y
Total: 6 150
Total credits of in-class work: 2.4 Total class time hours: 60
Total credits of out of class work: 3.6 Total hours of out of class work: 90

As: Assessable training activity
Com: Training activity of compulsory overcoming (It will be essential to overcome both continuous and non-continuous assessment).

8. Evaluation criteria and Grading System
Evaluation System Continuous assessment Non-continuous evaluation * Description
Assessment of active participation 20.00% 10.00% Active participation in solving problems
Assessment of activities done in the computer labs 30.00% 20.00% Resolution of practice exercises in the computer lab
Practicum and practical activities reports assessment 50.00% 70.00% Delivery of proposed works
Total: 100.00% 100.00%  
According to art. 6 of the UCLM Student Evaluation Regulations, it must be provided to students who cannot regularly attend face-to-face training activities the passing of the subject, having the right (art. 13.2) to be globally graded, in 2 annual calls per subject , an ordinary and an extraordinary one (evaluating 100% of the competences).

Evaluation criteria for the final exam:
  • Continuous assessment:
    Assessment of active participation, work in the computer lab, and autonomous work
  • Non-continuous evaluation:
    Evaluation criteria not defined

Specifications for the resit/retake exam:
Evaluation criteria not defined
Specifications for the second resit / retake exam:
Evaluation criteria not defined
9. Assignments, course calendar and important dates
Not related to the syllabus/contents
Hours hours

Unit 1 (de 5): Finite Differences
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 2 (de 5): Spectral Methods
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 3 (de 5): Finite Elements
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 4 (de 5): Finte Volumes and Finite Differences
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 5 (de 5): Courses and seminars
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 5.5
Computer room practice [PRESENCIAL][Problem solving and exercises] 1.5
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 10

Global activity
Activities hours
10. Bibliography and Sources
Author(s) Title Book/Journal Citv Publishing house ISBN Year Description Link Catálogo biblioteca
20. A.Quarterioni, R.Sacco, F.Saleri Numerical Mathematics Springer-Verlag 2000  
C. Bernardi and Y. Maday Approximations spectrales de problemes aux limites elliptiques Springer 1992  
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang Spectral Methods for Fluid Dynamics Springer 1988  
C. Johnson Numerical solution of P.D.E. by the Finite Element Method Cambridge University Press 1987  
D. Gotlieb and S. Orszag Numerical Analysis of Spectral Methods ¿Theory and Applications SIAM 1977  
D.F.Griffiths, A.R.Mitchell The finite difference method in partial differential equation John Wiley 1980  
E. Godlewski, P.A. Raviart Hyperbolic systems of conservation laws Ellipses 1991  
E.Godlewski, P.A. Raviart Numerical Approximation of Hyperbolic Systems of Conservation Laws Springer-Verlag 1996  
G.D. Smith Numerical Solution of Partial Differential Equations: Finite Difference Methods Oxford University Press 1985  
G.F.Forsythe, W.R.Wasow Finite difference methods for partial differential equations John Wiley 1960  
J. C. Strikwerda Finite difference Schemes and Partial Differential Pacific Grove, CA: Wadsworth and Brooks 1989  
J.H.Mathews, K.D. Fink Métodos Numéricos con MATLAB Prentice-Hall 2000  
J.M. Sanz-Serna Fourier techniques in numerical methods for evolutionary problems. 3RD Granada Seminar on Computational Physics Springer 1995  
L.N. Trefethen Spectral methods in Matlab SIAM 2000  
O.C. Zienkiewicz The Finite Element Method in Engineering Science McGraw-Hill 1971  
P.G.Ciarlet The finite element method for elliptic problems North Holland 1978  
R. LeVeque Finite Volume Methods for Hyperbolic Problems Cambridge Univesity Press 2002  
R.B. Richtmyer, K.W. Morton Difference methods for initial-value problems John Wiley & Sons 1967  
R.G. Voigt, D. Gotlieb and M.Y. Hussaini Spectral Methods for Partial Differential Equations SIAM 1984  
S. Nakamura Análisis Numérico y visualización gráfica con MATLAB Pearson Educación/Prentice-Hall Hispanoamerica 1997  



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