Summary of course and the final oral exam

Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no [1, 2]

[1] Department of Physics and Center of Mathematics for Applications, University of Oslo [2] National Superconducting Cyclotron Laboratory, Michigan State University

Spring 2018

Learning outcomes

Our ideal about knowledge on computational science

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.