Calculus, algebra, differential equations and functional analysis.
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.
Course competences | |
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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) |
Course learning outcomes | |
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Description | |
Interpretation of the obtained numerical solution and critical judgment of its quality. Relation with the applied science referred to | |
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 | |
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. | |
Additional outcomes | |
Not established. |
Training Activity | Methodology | Related Competences (only degrees before RD 822/2021) | ECTS | Hours | As | Com | Description | |
Class Attendance (theory) [ON-SITE] | Lectures | CB06 | 1.5 | 37.5 | Y | N | Theoretical development of the course contents and resolution of exercises | |
Computer room practice [ON-SITE] | Problem solving and exercises | CB07 CB08 CB09 CB10 CE01 CE02 CE05 | 0.7 | 17.5 | Y | Y | Troubleshooting in the computer room | |
Individual tutoring sessions [ON-SITE] | CT03 CT05 | 0.2 | 5 | Y | N | Individual tutorials for resolution of doubts | ||
Writing of reports or projects [OFF-SITE] | Self-study | CE07 CE08 CG03 CG04 | 3.6 | 90 | Y | Y | Resolution of problems and preparation of reports and works on the contents of the course | |
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).
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% |
Not related to the syllabus/contents | |
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Hours | hours |
Unit 1 (de 5): Finite Differences | |
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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 | |
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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 | |
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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 | |
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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 | |
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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 | |
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Activities | hours |
Author(s) | Title | Book/Journal | Citv | Publishing house | ISBN | Year | Description | Link | Catálogo biblioteca |
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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 | ||||||
C.H. Edwards, D. E. Penney | Ecuaciones diferenciales y problemas con valores en la frontera. Cómputo y modelado (4ª Ed) | Pearson | 2009 | ||||||
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. Romera, M.C. Boscá, F. Arias, F.J. Gálvez, J.I. Porras | Métodos Matemáticos: Problemas de Espacios de Hilbert, Operadores Lineales y Espectros | Paraninfo | 2013 | ||||||
E.Godlewski, P.A. Raviart | Numerical Approximation of Hyperbolic Systems of Conservation Laws | Springer-Verlag | 1996 | ||||||
Frederic Hecht | FreeFEM Documentation. Release 4.6 | 2021 | Pagina oficial de FreeFEM | https://freefem.org/ | |||||
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 | ||||||
Gwynne A. Evans, G. Evans, G. A. Evans, Jonathan M. Blackledge, J. Blackledge, Peter D. Yardley, P. Yardley | Numerical Methods for Partial Differential Equations | Springer | 2000 | ||||||
Haïm Brézis | Análisis Funcional | Alianza Universidad Textos | 84-206-8088-5 | 1983 | |||||
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 | ||||||
J.N. Reddy | An Introduction to the Finite Element Method | MCGRAW HILL SERIES IN MECHANICAL ENGINEERING | 2005 | ||||||
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 | ||||||
P.M. Gresho, R.L. Sani | Incompressible Flow and the Finite Element. Volume Two. Isothermal Laminar Flow | John Wiley & Sons | 0471492507 | 2000 | |||||
P.M. Gresho, R.L. Sani | Incompressible Flow and the Finite Element. Volume One. Advection-Diffusion | John Wiley & Sons | 0471492493 | 2000 | |||||
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 | ||||||
Randall LeVeque | Finite difference methods for ordinary and partial differential equations | SIAM | 2007 | ||||||
Roberto Font, Francisco Periago | The Finite Element Method with FreeFem++ for beginners | The Electronic Journal of Mathematics and Tecnology, Volume 7 (4) | 1933-2823 | 2013 | |||||
S. Nakamura | Análisis Numérico y visualización gráfica con MATLAB | Pearson Educación/Prentice-Hall Hispanoamerica | 1997 | ||||||
Sandip Mazumder | Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods | Elsevier Science | 2016 | ||||||
Stig Larsson, Vidar Thomee | Partial Differential Equations with Numerical Methods | Springer | 2014 |