• Blended Learning & Flipped Classroom

Course goals

The goal of the course is to provide students with the theoretical knowledge of the numerical techniques for the approximation of basic concepts in analysis and solutions of ordinary and partial differential equations, as well as with the ability to apply it to relevant model problems. The course covers the fundamental topics of numerical mathematics and the basic finite difference techniques for the approximation of ordinary and partial differential equations. All topics are presented at a theoretical level and the corresponding methods are applied to the solution of numerical problems in the computer laboratory sessions based on the use of MATLAB scientific software.

Course organization

The course will be organized according to the flipped classroom approach. Recorded lectures giving an introduction to the topics for each week will be provided in advance during the previous week, along with the exercises that will be discussed in the classroom. Simple automatic tests will also be provided for self assessment of the level of understanding of each topic. Study of the theoretical topics will have to be started before the meeting with the teacher devoted to the corresponding topic. During the meetings with the teacher, students will be asked basic questions on the theoretical topics to assess their level of understanding and to start the discussion on each topic. The course will be offered in either attendance or non attendance mode, to which will correspond different evaluation procedures. To be eligible for the attendance mode, students will be required to attend at least 75% of the classroom activities and to participate actively, according to the teacher's judgement, in the discussion on the theoretical topics and in the solution of exercises.

Attendance mode: Students who have attended at least 75% of the classroom activities and participated actively in the discussion on the theoretical topics and in the solution of exercises according to the teacher's judgement will be considered in attendance mode.

Non attendance mode: Students who attend less than 75% of the classroom activities or do not participate actively in the same activities will be
considered as taking the course in non attendance mode. All the material and the introductory recorded lectures will be available online and the teacher will be available two hours per week outside the lectures for assistance on the course topics and material.

The students will acquire detailed knowledge and understanding (DD1) of the basic principles of numerical approximation and will be able to apply this knowledge (DD2) to the solution of relevant numerical probems making use of the resources of the MATLAB scientific software. The student will be able to make judgements (DD3) on the most appropriate numerical method to be applied for the solution of different classes of problems in terms of accuracy and efficiency. The student will be able to present the results of the numerical approximation of the solution to a mathematical problem with appropriate technical language (DD4) and to understand mathematical proofs presented in applied mathematics textbooks. 

1) Introduction to basic concepts of numerical analysis and scientific computing.

2) Floating point representation of real numbers: Machine accuracy, cancellation of significant digits.

3) Polynomial interpolation: Existence and unicity of the interpolating polynomial. Bounds on the approximation error. Newton's form for the interpolating polynomial. Composite polynomial interpolation.

4) Methods for nonlinear equations: Bisection method: error estimate and convergence proof. Newton's method and its variants (chord, secant method). Fixed point method: sufficient conditions for convergence.

5) Finite difference approximation of derivatives: Forward, backward and centered finite differences. Finite difference approximation of second order derivatives.Richardson extrapolation.

6) Numerical integration methods: Basic quadrature rules: midpoint, trapezoidal and Simpson rule. Composite integration rules. Error estimates for simple and composite rules. Numerical computation of Fourier series coefficients (FFT).

7) Numerical methods for ordinary differential equations: Overview of basic existence and uniqueness theorems. Examples of simple numerical methods: forward Euler, Heun, second order Runge Kutta, leapfrog, backward Euler, Crank Nicolson, Adams-Bashforth, higher order Runge-Kutta. Convergence of one step methods. A-stability of numerical methods. Extension to ODE systems.

8) Numerical methods for linear systems: Methods for upper and lower triangular systems. Gaussian elimination and LU factorization. Pivoting. Special cases of gaussian elimination: tridiagonal systems. Singular value decomposition. Condition number of a matrix and error analysis of numerical methods for linear systems.

9) Application of Fourier series to the numerical solution of linear PDEs by separation of variables.

10) The linear, 1d advection-diffusion equation: Existence and uniqueness of solutions, representation formulae. Properties of the solution: regularity, maximum principle. Boundary conditions.

12) Finite difference methods for the linear, 1d advection diffusion equation: Examples of basic methods. Consistency, convergence and stability. Analysis of truncation error: numerical diffusion.

Students must have good theoretical knowledge of all the results on real and complex number theory, mathematical analysis of real and vector valued functions, analytic geometry, basic linear algebra theory and ordinary differential equations that are included in Engineering Bachelor courses at Politecnico di Milano.

Attendance mode: During the meetings with the teacher, students will be asked basic questions on the theoretical topics to assess their level of knowledge and understanding and their communications skills (DD1, DD4). These questions will start the discussion on each topic. Students will then be allowed to take two intermediate written tests (in November and December), consisting in some exercises to be solved with MATLAB, aimed at assessing the ability to apply knowledge and understanding, to make judgements and the communication skills in the presentation of the results (DD2,DD3,DD4).

Students who pass both intermediate tests will be given a final mark for the course before the December break.

Students who do not pass all the intermediate tests or want to improve their mark will be required to take the exam as in non attendance mode.

Non attendance mode: Students who attend less than 75% of the classroom activities or do not participate actively in the same activities according to the teacher's judgement will be considered as taking the course in non attendance mode. Students in non attendance mode cannot participate to the intermediate tests and will be required to take

1) a written test after the end of course, consisting in some exercises to be solved with MATLAB, aimed at assessing the ability to apply knowledge and understanding,the ability to make judgements and the communication skills in the presentation of the results (DD2,DD3,DD4).

2) an oral discussion on the exam results which will include questions on the theoretical topics of the course, aimed at assessing their level of knowledge and understanding and their communications skills (DD1, DD4).