## 27029 - Numerical Simulation in Ordinary Differential Equations

### Syllabus Information

2018/19
Subject:
27029 - Numerical Simulation in Ordinary Differential Equations
Faculty / School:
Degree:
453 - Degree in Mathematics
ECTS:
6.0
Year:
4
Semester:
First semester
Subject Type:
Optional
Module:
---

### 1.1. Aims of the course

This is an optional course in the degree of Mathematics. Its goal is to present the essentials of methods for the numerical solution of differential problems

### 1.2. Context and importance of this course in the degree

The subject belongs to the module "Cálculo Científico y Simulación Numérica". To take this subject

it is highly convenient to have passed the matters "Análisis matemático", "Ecuaciones Diferenciales", "Informatica",

"Análisis Numérico I" y "Análisis Numérico II.

### 1.3. Recommendations to take this course

The attendace to the class lectures and the computer laboratory sessions is highly recomended, as well as the
individual work on the problems posed along the course.
It is highly convenient to have passed the subjects "Análisis matemático", "Ecuaciones Diferenciales", "Informatica",

"Análisis Numérico I" y "Análisis Numérico II.

### 2.2. Learning goals

At the end of this course students should be able to:

• Know criteria to compare  and evaluate several numerical methods taking into account the computational cost.
• Evaluate the numerical results obtained and draw conclusions
• Know how approximate numerically the solution of an initial value problem and estimate the error committed by the numerical method.
• Know the limitations and advantages of the numerical methods under consideration.
• Know some commercial software (e.g. matlab, Mathematica, ...) and free software (e.g. ipython,...) for the numerical solution
• of differential problems.

### 3.1. Assessment tasks (description of tasks, marking system and assessment criteria)

As a general rule, the module can be passed either showing a regular work along the academic year, or by
a final exam.

• Regular work. During the course, the student results will be evaluated through a periodical supply of exercises or short tasks, together with their active participation during the course. The use of LaTeX in written presentations is recommended; the evaluation include an oral presentation using Beamer. These evaluations will constitute the final mark.
• Final exam. The aforementioned procedure does not exclude the right, according to the current regulations, to a final exam which, by itself, allows to pass the module.

### 4.1. Methodological overview

The methodology followed in this course is oriented towards the achievement of the learning objectives. A wide range of teaching and learning tasks are implemented, such as lectures, computer laboratory sessions, tutorials and autonomous work and study.

This course is organized as follows:

• Lectures. Theoretical results will be explained here.
• Computer laboratory sessions. Using software tools (ipython, ...) to illustrate applications of the theoretical results seen in lectures.
• Autonomous work and study. Study and completion of assignments.
• Tutorials. Teachers will attend students individually during office hours.
• Assessment tasks. Problems and homework tasks will be proposed during the course, allowing to pass the course. There will also be a final exam.

### 4.3. Syllabus

This course will address the following topics:

• Topic 1. One-step methods. Consistency, stability and convergence
• Topic 2. Linear multistep methods.
• Topic 3. Boundary Value Problems. Shooting methods
• Topic 4. Implementation of the numerical schemes and numerical simulation

### 4.4. Course planning and calendar

As a general rule, this course has four weekly face-to-face hours.

Further information concerning the timetable, classroom, office hours, assessment dates and other details regarding this course will be provided on the first day of class or please refer to the Faculty of Sciences website and Moodle.

### 4.5. Bibliography and recommended resources

• Hairer, Ernst. Solving ordinary differential equations. I, Nonstiff problems / E. Hairer, S.P. Nà¸rsett, G. Wanner. - 2nd rev. ed., 2nd corr. print. Berlin [etc.] : Springer-Verlag, 2000
• Hairer, Ernst. Solving ordinary differential equations. II, Stiff and differential-algebraic problems / E. Hairer, G. Wanner. - 2nd rev. ed, 2nd corr. print. Berlin [etc.] : Springer-Verlag, 2002
• Ascher, Uri M.. Numerical solution of boundary value problems for ordinary differential equations / Uri M. Ascher, Robert M.M. Mattheij, Robert D. Russell New Jersey : Prentice Hall, cop. 1988

## 27029 - Numerical Simulation in Ordinary Differential Equations

### Información del Plan Docente

2018/19
Subject:
27029 - Numerical Simulation in Ordinary Differential Equations
Faculty / School:
Degree:
453 - Degree in Mathematics
ECTS:
6.0
Year:
4
Semester:
First semester
Subject Type:
Optional
Module:
---

### 1.1. Aims of the course

This is an optional course in the degree of Mathematics. Its goal is to present the essentials of methods for the numerical solution of differential problems

### 1.2. Context and importance of this course in the degree

The subject belongs to the module "Cálculo Científico y Simulación Numérica". To take this subject

it is highly convenient to have passed the matters "Análisis matemático", "Ecuaciones Diferenciales", "Informatica",

"Análisis Numérico I" y "Análisis Numérico II.

### 1.3. Recommendations to take this course

The attendace to the class lectures and the computer laboratory sessions is highly recomended, as well as the
individual work on the problems posed along the course.
It is highly convenient to have passed the subjects "Análisis matemático", "Ecuaciones Diferenciales", "Informatica",

"Análisis Numérico I" y "Análisis Numérico II.

### 2.2. Learning goals

At the end of this course students should be able to:

• Know criteria to compare  and evaluate several numerical methods taking into account the computational cost.
• Evaluate the numerical results obtained and draw conclusions
• Know how approximate numerically the solution of an initial value problem and estimate the error committed by the numerical method.
• Know the limitations and advantages of the numerical methods under consideration.
• Know some commercial software (e.g. matlab, Mathematica, ...) and free software (e.g. ipython,...) for the numerical solution
• of differential problems.

### 3.1. Assessment tasks (description of tasks, marking system and assessment criteria)

As a general rule, the module can be passed either showing a regular work along the academic year, or by
a final exam.

• Regular work. During the course, the student results will be evaluated through a periodical supply of exercises or short tasks, together with their active participation during the course. The use of LaTeX in written presentations is recommended; the evaluation include an oral presentation using Beamer. These evaluations will constitute the final mark.
• Final exam. The aforementioned procedure does not exclude the right, according to the current regulations, to a final exam which, by itself, allows to pass the module.

### 4.1. Methodological overview

The methodology followed in this course is oriented towards the achievement of the learning objectives. A wide range of teaching and learning tasks are implemented, such as lectures, computer laboratory sessions, tutorials and autonomous work and study.

This course is organized as follows:

• Lectures. Theoretical results will be explained here.
• Computer laboratory sessions. Using software tools (ipython, ...) to illustrate applications of the theoretical results seen in lectures.
• Autonomous work and study. Study and completion of assignments.
• Tutorials. Teachers will attend students individually during office hours.
• Assessment tasks. Problems and homework tasks will be proposed during the course, allowing to pass the course. There will also be a final exam.

### 4.3. Syllabus

This course will address the following topics:

• Topic 1. One-step methods. Consistency, stability and convergence
• Topic 2. Linear multistep methods.
• Topic 3. Boundary Value Problems. Shooting methods
• Topic 4. Implementation of the numerical schemes and numerical simulation

### 4.4. Course planning and calendar

As a general rule, this course has four weekly face-to-face hours.

Further information concerning the timetable, classroom, office hours, assessment dates and other details regarding this course will be provided on the first day of class or please refer to the Faculty of Sciences website and Moodle.

### 4.5. Bibliography and recommended resources

• Hairer, Ernst. Solving ordinary differential equations. I, Nonstiff problems / E. Hairer, S.P. Nà¸rsett, G. Wanner. - 2nd rev. ed., 2nd corr. print. Berlin [etc.] : Springer-Verlag, 2000
• Hairer, Ernst. Solving ordinary differential equations. II, Stiff and differential-algebraic problems / E. Hairer, G. Wanner. - 2nd rev. ed, 2nd corr. print. Berlin [etc.] : Springer-Verlag, 2002
• Ascher, Uri M.. Numerical solution of boundary value problems for ordinary differential equations / Uri M. Ascher, Robert M.M. Mattheij, Robert D. Russell New Jersey : Prentice Hall, cop. 1988