33957 - Numerical Methods M

Learning outcomes

Knowledge of the fundamental algorithms for solving problems arising from mechanical engineering sciences, with particular attention to ordinary differential problems and partial differential equations.

Ability to choose the appropriate numerical method for concrete problems

Ability to interpret numerical results

Ability to implement numerical algorithms efficiently

Course contents

1- Introduction. Numerical Algorithms e Sources of Errors.

Numerical Algorithms, Errors, Algorithm properties. A very first introduction to conditioning and stability. Floating point systems and the IEEE standard system. Roundoff Errors.

2-Solving non-linear equations

Nonlinear equations in one variable, Bisection method, Fixed point iteration, secant method, Newton's method (and alternatives and variants). Newton method for nonlinear systems*

3-Solution of Linear Systems:

Crash Review on Linear Algebra. Direct methods for solving linear systems: Gauss-Elimination and LU factiorization. Solution of triangular systems. Pivoting. Cholesky decomposition. Errors and condition number. Simple Iterative methods (for linear systems): Jacobi and Gauss-Siedel.

4- Basics of Numerical Interpolation, Numerical Integration and Differentiation

Lagrange interpolation and Piecewise Lagrange interpolation

Interpolatory quadrature formulas: midpoint, trapezoidal, Simpson and error analysis. Divided Differences

5- Ordinary Differential Equations

Initial value ordinary differential equations. One Step methods: Euler’s method and Runge-Kutta methods. Absolute stability and stiffness. Richardson Extrapolation. Error control and estimation. Multistep methods* or Geometric integration*

6- Finite Differences for Partial Differential Equations

Transport equation, wave equation, heat equation. Finite Differences. Consistency, Stability and Convergence

7- Approximation of functions and data. (Least squares and Polynomial Interpolation).

Revisit Lagrange interpolation. Runge Phenomenon.Piecewise interpolation and Splines.
Least squares method for data fitting: linear regression and various examples.

Readings/Bibliography

- A First Course in Numerical Methods, Uri M. Ascher and Chen Greif, SIAM 2011

-Matematica Numerica, A. Quarteroni, R. Sacco, F. Saleri, Springer 2006

-MATLAB Guide, D. Higham, N. Higham,SIAM, 2000

Teaching methods

Lectures in class and in the Lab.

We will use MATLAB for all computer examples, exercises and projects.

Assessment methods

The exam for the course will be oral. During the course, several task-exercises will be proposed on the different topics. The elaboration of those exercises will be optional but will count toward the final vote. The students will be given the possibility of handing in several (selected) of those computer exercises-projects before the exam, and discuss them during the exam. 

Teaching tools

Different material will be provided during the course

Office hours

See the website of Blanca Ayuso