# Mathematical and Numerical Modelling of Conservation Laws (MOD600)

The course gives an introduction to the theory and applications of partial differential equations for modelling transport processes, with a particular focus on hyperbolic conservation laws. The structure of wave phenomena inherent in such models will be discussed, both for a scalar conservation law as well as systems. Furthermore, principles for the design and use of numerical methods to solve such models are presented. Examples of such models for different applications are also discussed.

Course description for study year 2023-2024. Please note that changes may occur.

Facts

MOD600

1

10

Spring

1

Spring

English

Time table

## 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.

## 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

1. 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;
2. be able to construct analytical solutions of certain classes of problems;
3. 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;
4. 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;
5. be able to implement in matlab/python a numerical scheme, compute and visualize approximate solutions.

## Required prerequisite knowledge

Need to meet the requirement for admission to the master program in Computational Engineering, Petroleum Engineering and other equivalent master engineering programmes.

## Recommended prerequisites

Good knowledge in mathematics (calculus) and physics. Some experience with coding (matlab/python) will be necessary. Basic knowledge of differential equations, numerical discretization and coding as presented in courses like MOD510/PET510 is recommended

## Coursework requirements

Mandatroy assignment (2-4)
Mandatory assignments (2-4) which must be approved 3 weeks before the exam.

Steinar Evje

## Course teacher:

Tore Halsne Flåtten

Øystein Arild

## Method of work

Class room instruction, programming exercises, calculation exercises

## Open for

Admission to Single Courses at the Faculty of Science and Technology Computational Engineering - Master of Science Degree Programme Petroleum Engineering - Master of Science Degree Programme Exchange programme at Faculty of Science and Technology

## Course assessment

There must be an early dialogue between the course coordinator, the student representative and the students. The purpose is feedback from the students for changes and adjustments in the course for the current semester.In addition, a digital course evaluation must be carried out at least every three years. Its purpose is to gather the students experiences with the course.

## Literature

The syllabus can be found in Leganto