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)