## Numerical Methods

This module at Master's level enables students to build mathematical models for common classes of engineering problems and solve the problems by implementing appropriately chosen numerical methods.

### Content

Topics include

- Basic concepts of mathematical modeling and numerical analysis
- Numerical differentiation and integration
- Numerical solutions of ordinary and partial differential equations
- Finite difference method (FDM), finite element method (FEM)
- Unconstrained and constrained optimization

The course description can also be found in the Campusboard-platform of the university.

### Learning goals

Students are able to

- explain basic concepts of mathematical modeling and numerical analysis
- describe numerical methods used in engineering with their respective advantages and limitations
- apply mathematical models to selected case studies
- estimate the errors inherent in different numerical methods
- implement numerical methods for different classes of problems using common software packages (e.g. MATLAB)
- develop custom solutions for non-standard use cases

### Literature

Here a selection of recommended readings.

- S. Chapra, R. Canale: Numerical Methods for Engineers. McGraw-Hill, New York, 2010.
- E. Walter: Numerical Methods and Optimization. Springer, 2014.
- François E. Cellier: Continuous System Modeling. New York, NY: Springer. Online available http://dx.doi.org/10.1007/978-1-4757-3922-0,1991
- K. Velten: Mathematical modeling and simulation. Introduction for scientists and engineers. Weinheim [Germany]: Wiley-VCH, 2009.
- J. Stoer, R. Bulirsch: Introduction to Numerical Analysis. Springer, Berlin 2002

### Software

As software, we use mainly MATLAB and Octave.

## Lab organization

During lab sessions you will implement numerical methods for different classes of problems and test them on engineering problems. Assignments are carried out using MATLAB and Octave, they are both theoretical and practical, for example:

**Questions and answers**, for example: "What is the idea of higher-order Runge-Kutta methods?", "What does it mean if an ODE is stiff?"- Small exercises that show how a method works
- Programming

Example: Write a MATLAB function that implements the central differences formula for calculating the derivative of a function and test it.

Example: Write a MATLAB program that solves a given heat diffusion problem. The expected output is a surface plot of the solution.

## Learning resources

Since the practical work is carried out in MATLAB, it is essential to learn basical MATLAB skills before the course starts or at latest during the first weeks of the course, using the following learning resources:

### MATLAB-Tutorial

The **MATLAB Tutorial** gives a brief introduction to the MATLAB language using as environment MATLAB 2020.
Basic MATLAB syntax (variables, operators, input, output, arrays, matrices, functions, plotting) is illustrated using small examples
that are saved as MATLAB scripts, for example test_hello.m, test_output.m.
Most of the examples will also run in the open source software Octave.

### MATLAB Quiz

Test your basic understanding of MATLAB Development Environment and language syntax. This quiz consists of six questions, each with three to four possible answers, and there may be multiple correct or incorrect answers. Check those answers that you think are correct. If you have answered at least 50% of the answers correctly, you are prepared to take MATLAB a step further, for example by taking the course "Numerical Methods". Have fun!