Computational Techniques

Course Objective:

The student will get familiarised with Linux operating system, basic data-visualisation softwares, and will learn how to simulate some simple statistical ensembles in computer, for example, Ising models, spatio-temporal dynamics of molecules and fluid systems.

Learning Outcome:

LO-1: The student will learn how to use command-line interface to write codes in open-source platforms. 

LO-2: The students will develop their own scripts for verifying statistics of different ensembles using Monte-Carlo method. 

LO-3: They will benchmark phase-transition phenomena using Molecular dynamics algorithms for structureless particles.

LO-4: They will be able to compare growth rates of fluid instabilities using different spatiotemporal discretisation techniques.

Course Content:

Unit I: Introduction to computer simulation

Setting up Linux environment, compilers, editors etc

Basic Linux commands

Working knowledge of shell-scripting

Revision of numerical methods

Simple user-graphics interface example of simulation

Unit II: Simulation of Statistical Ensembles

Basic idea of random sampling 

Importance sampling

Principle of detailed-balance

Metropolis algorithm

Algorithms for grid generation

Solving 2D Ising model

Monte-Carlo algorithm for statistical physics

Unit III: Algorithms for Ordinary Differential Equations

Introduction to ODE solvers

Error analysis

Comparison between ODE solvers

Symplectic algorithms

Boundary conditions

Minimum image convension and Ewald sum

Thermostats and Barostats

Integrated quantities for statistical analysis

Unit IV: Numerical Solution of Partial Differential Equations

Equations of Fluid Dynamics

Finite Difference algorithm

Finite Element algorithm

Finite Volume algorithm 

Spectral algorithms

Flux limiters

Pseudo-Spectral method

Boundary effects

Comparison of accuracy

Course Evaluation:

Internal Test 1  (25 Marks) 

Internal Test 2 (25 Marks)

Internal Test 3 (25 Marks)

Final Exam (50 Marks)

Final evaluation sheet will be prepared using the best TWO out of the three Internal Tests (25+25 = 50 Marks) + Final Exam (50 Marks).

Textbooks:

A Guide to Monte Carlo Simulations in Statistical Physics, by David P Landau and Kurt Binder; Cambridge (2009)

The Art of Molecular Dynamics Simulation, by D C Rapaport; Cambridge University Press (2011)

Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers, by David A Kopriva;  Springer (2009)