Numerical Methods in Engineering Sciences

a.y. 2014/15 - 2015/16 - 2016/2017 - 2017/2018

Organization

The course is divided in two parts, strictly related to one another:
1) theoretical classes (see "Program" below);
2) computer laboratory for the implementation of some of the numerical methods discussed in the classroom.

Program

  • Numerical solution of boundary value problems for Partial Differential Equations (Pde)


    • - Finite Difference method on a model problem in 1D.

    - Consistency and Stability - Lax's Theorem for convergence of a numerical scheme

    - Finite Element method on a model problem in 1D: Variational formulation, continuous piecewise linear finite element approximation, stability and convergence; construction of the final system and comparison with finite differences.

    - Finite Element method on a model problem in 2D: Functional spaces H1 and H10, energy norm; Variational Formulation, continuous piecewise linear finite element discretization on triangular meshes; Stability and convergence; Explicit computation of the elementary stiffness matrix and right-hand side; Assembling and solution of the final system.

    - Various examples of boundary value problems in 2D

  • Numerical solution of initial value problems for Ordinary Differential equations (Ode)

    - One-step methods: Euler backward and forward, Crank-Nicolson, Heun; Stability and A-stability, consistency, convergence and order of convergence.

    - Multistep Methods: general structure, consistency and stability conditions; Explicit and Implicit Adams methods.

    - Runge-Kutta methods: consistency and stability conditions; example of construction of an explicit RK-method (Hints on predictor-corrector methods).

    - Systems of Ordinary Differential Equations: stiff problems.

  • Reminders on Linear Algebra:
    • - norms for vectors and matrices, scalar product, eigenvalues and eigenvectors; matrices: positive definite, diagonally dominant, triangular, tridiagonal.

  • Numerical solution of linear systems: direct methods

    • - Stability analysis: condition number.
    - Gauss factorization LU; implementational issues and costs.
    - Symmetric, positive definite matrices: Cholesky factorization.
    - Tridiagonal matrices: factorization and costs.

  • Numerical solution of linear systems: iterative methods:

    • - Splitting-methods: Jacobi, Gauss-Seidel, relaxation; convergence analysis and implementational issues; stopping criteria.
    - Gradient and Conjugate Gradient methods (basic idea)

  • Solution of nonlinear equations/systems

    • - Nonlinear equations: bisection and Newton's methods. Convergence, order of convergence, stopping criteria.
    - Nonlinear systems of equations: Newton's method and variants.

  • Approximation of functions and data

    • - Lagrange interpolation: interpolation error, piecewise Lagrange interpolation, order of approximation in various norms.
    - Least squares method for data fitting: linear regression and various examples.

  • Numerical integration:

    • - Interpolatory quadrature formulas: midpoint, trapezoidal, Simpson and error analysis.
    - Extension to dimension 2 on rectangular domains. Quadrature formulas on triangular domains: barycenter, vertex, and midpoint of the edges.

    Reference book:
    A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics-2nd edition, Springer Series: Texts in Applied Mathematics, Vol. 37 (2007).

    Introduction to Matlab (by P. Antolin): (pdf file)

    Grading procedure


    The exam will be written. Each student will be offered a couple of questions on subjects developed in the classes and has one hour to answer.

    There are two typologies of exam:

    Basic exam: it consists in a couple of simple questions and/or exercises, intended to verify the capability of applying the numerical algorithms, without the need for a deep understanding. The maximum grade is 24/30.

    Advanced exam: it consists in a couple questions (more theoretical than in the basic exam), intended to verify comprehension of the subjects and not just a mere application of ready-to-use formulas. The answers must be articulated with a certain mathematical precision. The maximum grade is 30/30 cum laude.

    Oral exam is not compulsory. However, students who got a positive grade in the written parte (i.e., at least 18/30) might choose to take an oral exam to try to get a better grade if they think that their preparation is good enough. Needless to say, the oral exam can change the written grade in the positive, as well as in the negative direction. In particular, the minimal grade in the written part plus a poor oral part might end up in a failed exam.

    Note: for students that chose the basic exam in the written part the maximum grade obtainable can never exceed 24/30.