Computational Physics Lectures: Introduction to the course

Morten Hjorth-Jensen [1, 2]

[1] Department of Physics, University of Oslo [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

2018

Overview of first week

Wednesday (we have 1420 from 4pm to 6pm)
1. Presentation of the course, aims and content
2. Introduction to C++ programming and numerical precision. Discussion of first project
3. Numerical differentiation and loss of numerical precision (chapter 3 lecture notes)
Computer lab: Friday (we have 1420 from 4pm to 8pm)
1. The first two weeks we focus on simple programming tasks, start to look at project 1 and to set up the software Git and a repository at Github as well as Qt Creator as one possible IDE. This week we discuss how to set up Git and obtain a Github account.

Please take the survey. Hopefully this information will allow us to tailor the course to your interests and background.

Reading suggestions and exercises

Read sections 2.1-2.5 and 3.1-3.2 of lecture notes:

Introduction to C++ programming

Numerical precision and C++ programming (chapter 2 of lecture notes)

Numerical differentiation and loss of numerical precision (chapter 3 lecture notes)

Work on warm up exercise (exercise 3.1 in Lecture notes) to demonstrate several programming elements and/or start looking at project 1

Lectures and ComputerLab

Lectures: Wednesdays from 4pm to 5.45pm

Weekly reading assignments needed to solve projects.

First hour of each lab session may be used to discuss technicalities, address questions etc linked with projects.

Detailed lecture notes, exercises, all programs presented, projects etc can be found at the Github address of the course.

Computerlab: Fridays. We have reserved a time slot from 4pm to 8pm.

Weekly plans and all other information are the github address of the course.

Four projects, all have to be approved. The first project is pass/not passed only while the last three projects are graded and count 25% each of the final mark. The course ends with a final oral exam where you present a project of your choice. The final oral exam accounts for the remaining 25% of the final grade.

Course Format

Use version control like Git for repository and all your material.

C/C++ is the default programming language during lectures, but Fortran2008 and Python are also used. All source codes discussed during the lectures can be found at the github address of the course. We recommend either C/C++, Fortran2008 or Python as languages. If you are tempted to explore Julia, feel free to.

Topics covered in this course

Numerical precision and intro to C++ programming

Numerical derivation and integration

Random numbers and Monte Carlo integration

Monte Carlo methods in statistical physics

Quantum Monte Carlo methods

Linear algebra and eigenvalue problems

Non-linear equations and roots of polynomials

Ordinary differential equations

Partial differential equations (may not be covered)

Parallelization of codes

High-performance computing aspects and optimization of codes

Syllabus

Linear algebra and eigenvalue problems, chapters 6 and 7.

Know Gaussian elimination and LU decomposition

How to solve linear equations

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

Cholesky and tridiagonal matrix decomposition

Syllabus

Linear algebra and eigenvalue problems, chapters 6 and 7.

Householder's tridiagonalization technique and finding eigenvalues based on this

Jacobi's method for finding eigenvalues

Singular value decomposition

Cubic Spline interpolation

Syllabus

Numerical integration, standard methods and Monte Carlo methods (chapters 4 and 11).

Trapezoidal, rectangle and Simpson's rules

Gaussian quadrature, emphasis on Legendre polynomials, but you need to know about other polynomials as well.

Brute force Monte Carlo integration

Random numbers (simplest algo, ran0) and probability distribution functions, expectation values

Improved Monte Carlo integration and importance sampling.

Syllabus

Monte Carlo methods in physics (chapters 12, 13, and 14).

Random walks and Markov chains and relation with diffusion equation

Metropolis algorithm, detailed balance and ergodicity

Simple spin systems and phase transitions

Variational Monte Carlo

How to construct trial wave functions for quantum systems

Syllabus

Ordinary differential equations (chapters 8 and 9).

Euler's method and improved Euler's method, truncation errors

Runge Kutta methods, 2nd and 4th order, truncation errors

How to implement a second-order differential equation, both linear and non-linear. How to make your equations dimensionless.

Boundary value problems, shooting and matching method (chap 9).

Syllabus

Partial differential equations, chapter 10.

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. Their stability conditions.

Explicit, implicit and Crank-Nicolson schemes, and how to solve them.

How to compute the Laplacian in Poisson's equation.

How to solve the wave equation in one and two dimensions.

Overarching aims of this course

Develop a critical approach to all steps in a project, which methods are most relevant, which natural laws and physical processes are important. Sort out initial conditions and boundary conditions etc.

This means to teach you structured scientific computing, learn to structure a project.

A critical understanding of central mathematical algorithms and methods from numerical analysis. In particular their limits and stability criteria.

Always try to find good checks of your codes (like solutions on closed form)

To enable you to develop a critical view on the mathematical model and the physics.

Additional learning outcomes

has a thorough understanding of how computing is used to solve scientific problems

knows some central algorithms used in science

has knowledge of high-performance computing elements: memory usage, vectorization and parallel algorithms

understands approximation errors and what can go wrong with algorithms

has experience with programming in a compiled language (Fortran, C, C++)

has experience with debugging software

has experience with test frameworks and procedures

can critically evaluate results and errors

understands how to increase the efficiency of numerical algorithms and pertinent software

understands tools to make science reproducible and has a sound ethical approach to scientific problems

Is able to write a scientific report with software like Latex

Computing knowledge

Our ideal about knowledge on computational science

Hopefully this is not what you will feel towards the end of the semester!

And, there is nothing like a code which gives correct results!!

J. J. Barton and L. R. Nackman,*Scientific and Engineering C++*, Addison Wesley, 3rd edition 2000.

B. Stoustrup, The C++ programming language, Pearson, 1997.

An excellent text is Discovering Modern C++

D. Yang, C++ and Object-oriented Numeric Computing for Scientists and Engineers, Springer 2000.

And the C++ resource network provides great help.

The Fortran tutorial is also very useful.

And for Python programmers, see the textbook by Hans Petter Langtangen

Extremely useful tools, strongly recommended

and discussed at the lab sessions.

Git and a repository at Github, this and next week (and later weeks as well).

ipython notebook (Jupiter notebook, a great tool)

Qt Creator as one possible IDE for editing and mastering computational projects (for C++ codes, see webpage of course), discussed during the whole semester. You can however use other IDEs as well such as VisualC++.

Armadillo as a useful numerical library for C++, highly recommended, discussed in connection with LinAlgebra lectures

Unit tests, discussed throughout the whole semester

Piazza for discussions and teaching material