Pre-requisites: PHY- 104, 203, 206, 209

PART A

1. Review of Numerical Methods and the Solution of Ordinary Differential Equations

• Introduction to Numerical Computing
• Errors in Numerical Methods, Numerical Instabilities
• Discretization, integration and differentiation
• Disintegrations.
• Movement of particles in various fields of forces.
• Oscillations, resonances, chaos.
• Gravitational problem with 𝑵 body.
• Movement of rigid solids
• Numerical Methods for Solving Ordinary Differential Equations
  • Euler’s Method
  • Higher-Order Methods
  • The Leap-Frog Method
  • The Runge-Kutta Method
  • The Predictor Corrector Method
  • The Intrinsic Method
• Stability analysis of numerical schemes
• Implementation of Numerical Methods in dynamic high-level scripting programming languages
• Adaptive Integration Methods
• Advanced Integration Methods
• The Chaotic Pendulum
• Phase Space
• Conserved Quantities
• Analytic Solution
• Numerical Solution
• Sources of Simulation Error
• The Poincaré Section
• Spatial Symmetry Breaking
• The Fast Fourier Transform and applications
• Map Based Schemes
• Introduction to Nonlinear Systems –
  • Period-Doubling Bifurcations,
  • The Route to Chaos,
  • Sensitivity to Initial Conditions,
  • The Definition of Chaos,
  • Periodic Windows

2. Partial Differential Equations

• Types of Equations
• Elliptic Equations –
  • Laplace's Equation
• Hyperbolic Equations –
  • Wave Equations,
  • wave propagation,
  • initial conditions and limits, modes,
  • Eigen frequencies,
  • stability analysis
• The 𝟏-𝑫 Advection Equation,
• The Lax Scheme,
• The Crank-Nicholson Scheme,
• Upwind Differencing,
• The 𝟏-𝑫 Wave Equation,
• The 𝟐-𝑫 Resonant Cavity
• Waves in inhomogeneous medium
• Eulerian and Lagrangian Methods
• Parabolic Equations Diffusion
• The Diffusion Equation –
  • 𝟏-𝑫 Problem With Dirichlet,
  • Neumann, Mixed Boundary Conditions,
  • Explicit schemes, finite differences.
  • Von Neumann Stability Analysis,
  • 𝟏-𝑫, 𝟐-𝑫 And 𝟑-𝑫 Solution Of the Diffusion Equation
• Courant-Friedrichs-Lewy Condition
• Conservative Methods –
  • The Equation Of Continuity
  • Maxwell's Equations
• Dispersion.

3. Matrix Algebra

• Introduction
• Types of Matrices
• Simple Matrix Problems
• Elliptic Equations –
  • Poisson's Equation,
  • 𝟏-𝑫 Problem with Dirichlet Boundary Conditions,
  • 𝟐-𝑫 Problem with Neumann Boundary Conditions,
  • Example Solution of Poisson's Equation in 𝟏-𝑫, 𝟐-𝑫 And 𝟑-𝑫
• Two-dimensional case: equations of electrostatics and magnetostatics
• Finite Difference and Finite Element Schemes
• Systems of Equations and Matrix Inversion –
  • Exact Methods,
  • Iterative Methods,
  • The Jacobi Method,
  • The Gauss-Seidel Method
• Matrix Eigenvalue Problems –
  • Schrödinger's Equation,
  • General Principles,
  • Full Diagonalization,
  • The Generalized Eigenvalue Problem,
  • Partial Diagonalization,
  • Sturm Sequence,
  • Sparse Matrices and The Lanczos Algorithm,
  • Numerical Solution of Schrodinger Equation for Spherically Symmetric Potentials –
    • Scattering States,
    • Calculation of Phase Shifts,
    • Resonance.

PART B

4. The Monte Carlo Methods and Simulation

• Monte Carlo -
  • Random Number Generators,
  • Distribution Functions
• Monte-Carlo Integration
• Diffusion and random walk
• Stochastic Processes, Markov Chain
• The Metropolis Algorithm –
  • The Ising Model,
  • Thermodynamic Averages
• Quantum Monte-Carlo
• Molecular Dynamics –
  • Interacting Particles with Lennard-Jones Potentials,
  • Classical and Tight Binding Molecular Dynamics,
  • Simulation of 𝑨𝒓
• Introduction to Particle Transport Simulation – Cross-Sections
• Simulation of Neutron Transport and Scattering – Nuclear Criticality with Monte Carlo.

5. Computer Applications

• Particle-In-Cell Codes –
  • Introduction,
  • Normalization Scheme,
  • Solution of Electron Equations of Motion,
  • Evaluation of Electron Number Density,
    • An Example 𝟏𝐃 𝐏𝐈𝐂 𝐂𝐨𝐝𝐞
• The theory of quantum mechanics.
  • From many-body to single-particle: Quantum modeling of molecules
• Density Functional Theory, Car-Parrinello Simulation
• Hubbard Model –
  • Motivation,
  • Representation of 𝑺𝒛 Basis,
  • Generation of Basis States,
  • Construction of Hamiltonian.
  • Exact Diagonalization,
  • Calculation of Correlation Function.
  • Lanczos Method and Applications to Tight Binding Hamiltonians,
  • Calculation of Spectral Properties.
• Quantum modeling of solids: Basic properties.
• Electrons in Periodic Potential,
• Calculation of Band Structure using Plane Wave Methods.