# Summary of course and the final oral exam **Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no**, Department of Physics and Center of Mathematics for Applications, University of Oslo and National Superconducting Cyclotron Laboratory, Michigan State University Date: **Spring 2018** ## Learning outcomes [Our ideal about knowledge on computational science](http://hplgit.github.io/edu/py_vs_m/computing_competence.html) Does that match the experiences you have made this semester?

## Topics we have covered this year

• Introduction to c++ programming and object orientation

• How to write scientific reports

• Linear algebra and eigenvalue problems. (Lecture notes chapters 6.1-6.5 and 7.1-7.5 and projects 1 and 2).

• Ordinary differential equations (Lecture notes chapter 8 and projects 3 and 4 )

• Monte Carlo methods in physics (Lecture notes chapters 11, 12 and 13, project 4)

## Linear algebra and eigenvalue problems, chapters 6.1-6.5 and 7.1-7.5

• How to handle vectors and matrices

• Gaussian elimination and LU decomposition (project 1)

• How to solve linear equations (project 1)

• How to obtain the inverse and the determinant of a real symmetric matrix

• Cubic spline

• Tridiagonal matrix decomposition (project 1)

• Householder's tridiagonalization technique and finding eigenvalues based on this

• Jacobi's method for finding eigenvalues (project 2)

## Monte Carlo methods in physics (Chapters 11, 12 and 13)

• Random walks and Markov chains

• Generation of random numbers

• Monte Carlo integration

• Probability distributions and their properties

• Errors in Monte Carlo calculations (statistical errors)

• Metropolis algorithm (project 4)

• Applications to statistical physics systems (project 4)

• Brief excursion into quantum mechanical systems (project 4)

## Ordinary differential equations (Chapter 8)

• Euler's method and improved Euler's method, truncation errors (projects 3 and 4)

• Runge Kutta methods, 2nd and 4th order, truncation errors (projects 3 and 4)

• Leap-frog and Verlet algoritm (projects 3 and 4)

• How to implement and solve a second-order differential equation, both linear and non-linear.

• How to make your equations dimensionless.

## Partial differential equations, chapter 10, not covered this year

• Set up diffusion, Poisson and wave equations up to 2 spatial dimensions and time

• Set up the mathematical model and algorithms for these equations, with boundary and initial conditions. The stability conditions for the diffusion equation.

• Explicit, implicit and Crank-Nicolson schemes, and how to solve them. Remember that they result in triangular matrices (project 4).

• Diffusion equation in two dimensions.

## Final presentation

Select the project you liked the most among projects 2-4. Alternatively, if there are other topics of relevance you would like to present, feel free to suggest (send me an email however). Your presentation (bring your own laptop) should include

• Introduction with motivation

• Give an overview of the theory and numerical algorithms employed

• Discuss the implementation of your algorithm and how you verified it and validated it. Discuss for example various tests you made.

• Present and discuss your results

• Summary, conclusions and perspectives

• Anything else you think is important. Useful to have backup slides

In total your talk should have a duration of 20-25 minutes, but longer is also ok. The style for the final presentation follows very much the layout your reports.