### Learning outcome

**Knowledge:**

The student will have an extended understanding of the following concepts in the context of an important class of partial differential equations:

1) Nonlinear and linear conservation laws;

2) Eigenvalue analysis and characteristics;

3) Analytical solutions and weak solutions;

4) Numerical discretization techniques;

5) Stability and accuracy of numerical approximations of the continuous model;

6) Insight into the connection between mathematical models for transport processes and

physical wave phenomena;

7) Practical coding experience through project work.

**Skills:**

The student will be able to

- understand how nonlinear and linear conservation laws naturally occur in the modeling of different dynamic processes that vary in space and time, man-made as well as phenomenon found in nature;
- be able to construct analytical solutions of certain classes of problems;
- formulate discrete schemes that can be used to compute numerical solutions of general conservation laws, both in the case of a single conservation law and a system of conservation laws;
- have an understanding of the challenges with finding solutions of nonlinear conservation laws and the need of concepts like shock wave and rarefaction wave, crossing characteristics, Rankine-Hugoniot jump condition, Riemann problem, entropy solution, and weak solutions;
- be able to implement in matlab/python a numerical scheme, compute and visualize approximate solutions.

### Contents

**Content**

Fundamental mathematical models for studying transport in continuous media, of general relevance for computational fluids engineering, chemical engineering, and biomedical engineering. Basic analysis of the wave structure inherent in the models as well as numerical discretization techniques are covered. Coding exercises/projects will be given on practical cases. Such models have a wide range of applications within fluid mechanics, chemical engineering, earth science, biomedical engineering, and material science, to mention some examples.

### Required prerequisite knowledge

### Recommended previous knowledge

Good knowledge in mathematics (calculus) and physics. Some experience with coding (matlab/python) will be necessary.

### Exam

Weight | Duration | Marks | Aid | |
---|---|---|---|---|

Written exam | 1/1 | 4 hours | A - F | Standard calculator. |

### Course teacher(s)

- Course coordinator
- Steinar Evje
- Course teacher
- Tore Halsne Flåtten

### Method of work

### Open to

Computational Engineering, Master's Degree Programme

Petroleum Engineering - Master of Science Degree Programme

Petroleum Engineering - Master`s Degree programme in Petroleum Engineering, 5 years

### Course assessment

### Literature

Sist oppdatert: 05.06.2020