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 2021-2022

Facts
Course code

MOD600

Version

1

Credits (ECTS)

10

Semester tution start

Spring

Number of semesters

1

Exam semester

Spring

Language of instruction

English

Time table
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.
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
Need to meet the requirement for admission to the master programme 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
Exam
Coursework requirements
Mandatroy assignment (2-4)
Mandatory assignments (2-4) which must be approved 3 weeks before the exam.
Course teacher(s)
Course coordinator: Steinar Evje
Course teacher: Tore Halsne Flåtten