2025
Numerical Analysis II
Name: Numerical Analysis II
Code: MAT14225L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Presentation
It is intended that the student performs a systematic study of numerical methods for solving ordinary differential equations (ODE) as well as an elementary study of numerical methods for problems with partial derivatives. Implementation of some methods using free software.
Sustainable Development Goals
Learning Goals
Systematic study of numerical methods for solving ordinary differential equations (ODE) and an elementary introduction to numerical methods for problems with partial derivatives. Implementation of some methods using free software (Python, Jupyter or SageMath).
In addition to the use of computational tools, it is intended to acquire theoretical mathematical results that allow the development of generalization and abstraction skills, which, in turn, will allow formulating and solving problems efficiently. With the preparation of a computational project throughout the semester, it is intended that students use mathematical models, acquire the ability to work / learn autonomously, with clear and rigorous writing and above all a critical spirit
In addition to the use of computational tools, it is intended to acquire theoretical mathematical results that allow the development of generalization and abstraction skills, which, in turn, will allow formulating and solving problems efficiently. With the preparation of a computational project throughout the semester, it is intended that students use mathematical models, acquire the ability to work / learn autonomously, with clear and rigorous writing and above all a critical spirit
Contents
1. Ordinary differential equations: initial value problems.
- Single step methods. Euler's methods. Truncation error and consistency. Taylor's methods. Runge-Kutta methods. Convergence, Stability.
- Multiple step methods. Adams, Nystrom and Milne methods. Multi-step linear methods. Consistency, convergence, stability. Predictor-corrector process.
- Systems of ordinary differential equations. Stiffness: a brief reference. Ordinary differential equations of higher order.
2. Ordinary differential equations: boundary-values problems.
- Collocation method. Least squares method.
- Finite difference method. Errors and convergence.
- Weak symmetrical formulation.
- Introduction to the finite element method. Basis functions.
- Single step methods. Euler's methods. Truncation error and consistency. Taylor's methods. Runge-Kutta methods. Convergence, Stability.
- Multiple step methods. Adams, Nystrom and Milne methods. Multi-step linear methods. Consistency, convergence, stability. Predictor-corrector process.
- Systems of ordinary differential equations. Stiffness: a brief reference. Ordinary differential equations of higher order.
2. Ordinary differential equations: boundary-values problems.
- Collocation method. Least squares method.
- Finite difference method. Errors and convergence.
- Weak symmetrical formulation.
- Introduction to the finite element method. Basis functions.
Teaching Methods
Theoretical lectures, examples with emphasis on applications, resolution of exercises, practical work in computational laboratory, one computational project. The computational project is mandatory, carried out individually or in small groups.
Assessment
There will be two in-class assessments (F1 and F2) and examinations (regular, resit, special and extraordinary).
In addition to the in-class assessments and examinations, a computing project will be carried out in the form of an individual Practical Laboratory Test (PLT) completed on a computer in the classroom.
The PLT will be held on a single occasion and the mark obtained will be taken into account in any assessment scheme.
For the continuous assessment scheme, the marks from the first and second assessments and the PLT will be taken into account, with the final mark calculated according to the formula:
(2xF1 + 2xF2 + TPL)/5 (*)
In none of the assessments may the mark be less than 7.5 marks, and the average must be greater than or equal to 9.5 marks.
To pass under the continuous assessment system, the mark obtained using formula (*),
rounded to the nearest whole number, must be equal to or greater than 10 marks.
The mark for any examination (Ex) will be calculated using the formula,
(4 × Ex + TPL) / 5 (**)
where Ex must be equal to or greater than 9.5.
Formula (**) will apply to all types of examination: regular, resit, special or extraordinary.
A student will pass the course unit when the marks obtained using formulas (*) or (**), rounded to the nearest whole number, are 10 or above.
In addition to the in-class assessments and examinations, a computing project will be carried out in the form of an individual Practical Laboratory Test (PLT) completed on a computer in the classroom.
The PLT will be held on a single occasion and the mark obtained will be taken into account in any assessment scheme.
For the continuous assessment scheme, the marks from the first and second assessments and the PLT will be taken into account, with the final mark calculated according to the formula:
(2xF1 + 2xF2 + TPL)/5 (*)
In none of the assessments may the mark be less than 7.5 marks, and the average must be greater than or equal to 9.5 marks.
To pass under the continuous assessment system, the mark obtained using formula (*),
rounded to the nearest whole number, must be equal to or greater than 10 marks.
The mark for any examination (Ex) will be calculated using the formula,
(4 × Ex + TPL) / 5 (**)
where Ex must be equal to or greater than 9.5.
Formula (**) will apply to all types of examination: regular, resit, special or extraordinary.
A student will pass the course unit when the marks obtained using formulas (*) or (**), rounded to the nearest whole number, are 10 or above.
Teaching Staff
- Marília da Conceição Valente Oliveira Pires [responsible]
Accreditations and Certifications
