Numerical methods
[FSAB1104]
[30h + 30h] 5 credits
This course is taught in the 1st semester.
Teacher(s): Vincent Legat(coord.), Grégoire Winckelmans
Language: French
Level: First
Aims
The course serves three main purposes:
(1) the understanding of basic numerical techniques with the underlying mathematical notions,
(2) the maturity to interpret the reliability of numerical results,
(3) the programming skills to implement simple numerical algorithms.
At the end of the lecture, the student must be able to:
- identify physical reality, mathematical model and numerical solution,
- understand the numerical methods in terms of accuracy, convergence and stability,
- select a numerical methods taking into account accuracy and cost requirements,
- implement a numerical method in computer software,
- interpret and to validate the computed results, which may lead to further refinement of the mathematical model
The goal is to cover a wide range of numerical methods to obtain an approximate solution of problems of physics where an exact solution is not available. A broad knowledge is often decisive to choose the right method when developing a new code. A strong emphasis is put on the problem based learning where the participants analyze data, derive, implement, document and execute their own models. Finally, the analytical and the numerical approaches are presented as complementary tools.
Main themes
Topics include
- Error analysis: modelling error, truncation error, convergence and approximation order, floating point number representation (IEEE754).
- Approximation and interpolation: Lagrange polynomials, spline functions, NURBS, orthogonal polynomials, error estimators.
- Numerical integration and differentiation: backward and centered finite difference, midpoint, trapezoidal and simpson formula, adaptive techniques.
- Ordinary Differential Equations (ODE): Taylor and Runge Kutta methods, predictor-corrector methods, stability on unbounded intervals and perturbation analysis.
- Linear equations: factorization methods and iterative techniques, complexity, computation of eigenvalues.
- Nonlinear equations: bisection and Newton methods, optimisation applications.
- Partial Differential Equations (PDE): boundary value problems (Laplace, heat equation, waves equation), approximation by finite differences.
Although numerous concrete applications, the student will acquire a working knowledge in numerical methods using a problem-based learning environment;
Content and teaching methods
Other information (prerequisite, evaluation (assessment methods), course materials recommended readings, ...)
- Workfiles for each of the parts (available on the website and in