Computational Methods (PH41012)

Semester: Spring 2020-21

Syllabus, references and grading policy

This course is divided in two components: (1) Theory lectures (2 Credits); (2) Lab work (2 Credits).

Prerequisite: Motivation and willingness to learn. Rest is secondary.

Instructor (Theory): Vishwanath Shukla

Instructors (Laboratory): Vishwanath Shukla and Jyotirmoy Bhattacharya 

Teaching Assistant: Sudip Das, Sayan Das , Pritam Das, Amit Kumar Pradhan, Sudipta Biswas, Anuj Pratim Lara 

Lecture Notes

Presentation Slides

Syllabus

Number representation: Floating-point representation; Roundoff error; truncation error; stability.

Solution of linear algebraic equations: Gauss-Jordan elimination; LU decomposition.

Random numbers: Pseudorandom numbers; Random number generation algorithms; Testing randomness; Uniform distribution; Gaussian Distribution; Random walk simulation.

Numerical differentiation and integration: Finite difference schemes to approximate derivatives (Forward and Central difference); Error assessment. Integration: Quadrature as box counting (numerical quadrature); Trapezoidal rule; Simpson’s rule; Gaussian quadrature; Monte Carlo integration; Importance sampling method; von Neuman rejection method.

Root finding: Bracketing and bisection method; Newton-Raphson method; Roots of polynomials; Newton-Raphson method for nonlinear systems of equations.

Data fitting: Extrapolation; Linear interpolations; Lagrange interpolation; Cubic-spline interpolations; Leas-Squares fitting; Linear quadratic fit; Nonlinear fit.

Ordinary and partial differential equations: Euler method, Runge-Kutta (RK2, RK4, RK45) method, Adams-Bashforth Predictor-Corrector method.

Eigenvalue problems using ODE solver and bisection.

Two-point boundary value problems: Shooting method.

Elementary ideas of numerical solution of partial differential equations.

Fast Fourier transform: Fourier transform of discretely sampled data, Fast Fourier transform, Convolution, correlation and autocorrelation using FFT, Power spectrum estimation using FFT. 

Statistical description of data: Moments of a distribution (Mean, Variance, Skewness, etc).

Elementary ideas of parallel computing. Supercomputers. (3)

Examples and applications: Logistic maps; Fixed points; Period doubling; Attractors; Chaotic pendulum; Fractals; Computing fractal dimensions; Time-independent Schrödinger equation; Quantum mechanical scattering problems; Heat equation; Wave equation; Laplace equation; Poisson equation. 

References

(Below is a representative list. We will discuss about it later in the classroom.)

  1. R. H. Landau, M.J. Páez, C. C. Bordeianu. Computational Physics: Problems Solving with Computers. Wiley VCH.
  2. S. E. Koonin and D. C. Meredith. Computational Physics (Fortran Version). Westview Press. Advanced Book Program. 
  3. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanner. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.

Grading Policy: Continuous evaluation mechanism, as suggested by the Institute. 

Below is a tentative list of activities planned for the theory component.

  1. Participation in the (online) classroom activities; 
  2. Assignments to be solved and submitted during lab sessions;
  3. Quizzes/Viva during lab sessions; 
  4. Theory: (i) One compulsory test; (ii) Choice of a test or a mini project (as agreed upon);
  5. Bonus marks problems, if anyone is interested. 

Freedom to choose your own topics. 

For any other information please feel free to contact me.