• Blended Learning & Flipped Classroom

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 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.

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. For the first part of the course, the learning process will be organized around specific, application oriented goals in the framework of a flipped classroom mode. This interactive approach will guarantee greater direct participation of the students and allow for a smoother advancement towards the more complex topics in the second part of the course.

1) Introduction to basic concepts of numerical analysis: Approximation, relative error, conditioning of a numerical problem, review of analysis concepts ( Lagrange midpoint theorem, Taylor formula in one and more dimensions, order of infinitesimal).  

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

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) The Poisson equation and elliptic problems in one spatial dimension. Finite difference numerical methods for the approximation of elliptic problems in one spatial dimension.

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

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

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 usually included in Engineering Bachelor courses at Politecnico di Milano.

The assessment will consist in an individual written assessment and in an individual oral interview. In the written assessment, students will be first asked a few questions on the theoretical concepts presented in the course. They will then be assigned a set of numerical approximation problems, to be solved with the help of the MATLAB scientific software. They will be required to summarize and interpret the obtained results. The oral interview will focus on the discussion on the interpretation of the results and on the underlying theoretical concepts.  By solving the assigned problems and presenting correctly the results, the students will demonstrate a) the acquired ability (DD2)  to apply numerical methods for the solution of relevant analysis problems, with special focus on ordinary and partial differential equations b) the ability 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 c) the ability to present (DD4) the results of the numerical analysis of a differential problem with appropriate technical language.