Teaching plan for the course unit

 General information

Course unit name: Simulation Methods

Course unit code: 568180

Coordinator: Angel Jorba Monte

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 Estimated learning time Total number of hours 150

 Face-to-face learning activities 60
 -  Lecture 60
 Supervised project 45
 Independent learning 45

 Recommendations

 Students must have a good knowledge of numerical calculus and be able to complete efficient programs in a high-level language. It is highly advisable to know the theory of Ordinary Differential Equations and it would also be desirable to have some knowledge of Partial Differential Equations.

 Competences to be gained during study

 — Capacity to understand the mathematical foundations of simulation methods, their underlying geometry and analysis.   — Capacity to design, implement and test numerical methods for simulation: Analysis of the method, implementation issues concerning accuracy and efficiency.   — Capacity to interpret the obtained results: Their numerical aspects and those concerning the physical and/or geometrical meaning of the solutions found.

 Learning objectives
 Referring to knowledge Referring to knowledge — To know the most efficient and accurate methods to integrate ordinary differential equations, both initial value problems and boundary value problems, and partial differential equations of different kinds and with different degrees of accuracy. — To learn to use these methods to answer a variety of questions concerning: Long time evolutions, detection of regularity and chaos, detection of stability/instability, computation of periodic solutions, computation of invariant manifolds and their connections (homoclinic and heteroclinic orbits), problems related to control, effects of instability, effects of round off errors in regular orbits. — To be able to follow up on different kinds of solutions with respect to parameters and to analyze bifurcations.   Referring to abilities — To acquire the capability to select the suitable methods depending on the problem and on the requirements posed to the solution. — To acquire the capability to analyze the local truncation errors of the method and the numerical stability due to the propagation of errors. — To construct accurate and efficient programs in a high-level language and to discuss the validity of the results obtained and their analytical interpretation of the method, as well as implementation issues concerning accuracy and efficiency.

 Teaching blocks

No..

Title

1

 Integration methods for initial value problems for ODEs

2

 Applications to detect regularity/chaos, stability/instability, to compute periodic orbits and other invariant objects

3

 Continuation of invariant objects and analysis of bifurcations

4

 Finite difference methods for PDEs: Partial discretization and time-space discretizations

5

 Variational methods and finite elements for PDE

6

 Introduction to the effect of random terms and to Montecarlo methods

 Teaching methods and general organization

 Lectures will be mainly devoted to present the theoretical contents of the topics and to discuss the implementation aspects, how to check the validity of results and how to detect all kinds of errors. Independent learning must be devoted to understanding the principles of the methods and to develop the capabilities to analyze and implement them. The bulk of the course is the work done under supervision. Students must remember that each topic requires understanding the foundations as well as the solutions. In other words, they must acquire a good global knowledge of the different issues.

 Official assessment of learning outcomes

 To succeed in the assessment of the subject, students must show a good understanding of the foundations of the methods presented in the lectures, and a good ability to solve concrete problems. Problem-solving is essential for success. During the course, a number of questions will be asked to the students to be answered by doing the corresponding programs and presenting a detailed report on the method used and its implementation, the results obtained and the interpretation of the results at the light of the theory. The final grade will be calculated on the assessment of the solutions found to the proposed problems and the interpretation of the results.     Examination-based assessment Students desiring a single assessment process should inform the lecturer during the first two weeks of class. They will be asked to solve a number of theoretical questions in an examination (worth 1/3 of the final grade) and to solve some practical problems, including programs and numerical results, in a separate examination. The latter will be worth 2/3 of the final grade.