Numerical Methods & Computational Techniques

This is a 4 credit elective course for MSc second semester in Physics.

Course Outcome:

The students are expected to become skilled, both theoretically and practically, in computer programming and are expected to be able to solve numerical problems that are frequently used in physics using computer programs.

Course Content:

Unit I:

Solution of Equations in One Variable

Bisection, Fixed-Point Iteration, Newton-Raphson, Secant, Regula Falsi, Aitken's and Steffensen's Method for Accelerating Convergence, Horner's Method for Polynomial Derivatives, Muller's Method for Complex Zeros.

Interpolation & Polynomial Approximation 

Interpolation and Lagrange Polynomial: Neville's Method, Divided Differences, Hermite Interpolation, Natural and Clamped Cubic Spline Interpolation, Bezier Method for Parametric Curves.

Differentiation and Integration 

Richardson's Extrapolaton, Composite Numerical Integration, Romberg Integration, Adaptive Quadrature Method, Gaussian Quadrature, Multiple Integrals: Simpson and Gaussian Double Integral, Gaussian Triple Integral, Improper Integrals.

Unit II:

Initial-Value Problems for ODE

Euler's Method, Higher-Order Taylor Methods, Runge-Kutta Methods, Error Control and Runge-Kutta-Fehlberg Method, Multistep Methods: Adams Predictor-Corrector Method, Variable Step-Size Multistep Methods, Extrapolation Methods, Higher-Order Equations and Systems of Differential Equations, Stability, Stiff Differential Equations: Trapezoidal with Newton's Iteration.

Direct Methods for Solving Linear Systems 

Linear Systems of Equations: Gaussian Elimination with Backward Substitution, Gaussian Elimination with Partial Pivoting and Scaled Partial Pivoting, Determinant, Matrix Inversion and Factorization (LU, LDL^t, Cholesky and Crout for tridiagonal linear systems), Special Matrices.

Iterative Techniques in Matrix Algebra 

Jacobi and Gauss-Siedel Iterative Technique, Relaxation Techniques for Solving Linear Systems: Successive Over Relaxation (SOR), Error Bounds and Iterative Refinement, Preconditioned Conjugate Gradient Method.

Unit III:

Approximation Theory 

Discrete Least Square Approximation, Orthogonal Polynomials and Lease Square Approximation, Chebyshev Polynomials and Economization of Power Series, Rational Function Approximation: Pade, Chebyshev Rational Approximation, Trigonometric Polynomial Approximation, Fast Fourier Transforms.

Approximating Eigenvalues 

Orthogonal Matrices and Similarity Transformations, Power Method, Symmetric and Inverse Power Method, Wielandt Deflation, Householder's Method, QR Algorithm, Singular Value Decomposition.

Nonlinear Systems of Equations 

Newton's Method for Systems, Quasi-Newton Method: Broyden Method, Steepest Descent Techniques, Homotopy and Continuation Methods.

Unit IV:

Boundary-Value Problems for ODE 

Linear Shooting Method, Shooting Method for Nonlinear Problems, Finite-Difference Method for Linear and Nonlinear Problems, Rayleigh-Ritz Method.

Partial Differential Equations

Elliptic, Parabolic, Hyperbolic PDEs, Finite Element Method, Finite Volume Method, Shock Capturing Schemes: (TVD, MUSCL), WENO and Galerkin Algorithms, Implementation of Flux Limiters.

Course Evaluation:

Internal Test 1  (25 Marks): Hands-on implementation of algorithms in class and submission of assignments in time.

Internal Test 2 (25 Marks): Presentation on a specific assigned topic to individual students .

Internal Test 3 (25 Marks): Surprise Test

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).

Text Books:

Numerical Analysis, by R L Burden and J D Faires (Brooks/Cole, Cengage Learning)

Reference Books:

Fundamentals of Engineering Numerical Analysis, by P Moin (Cambridge Univ Press)