Course outline:
1. Interpolation (3 hrs)
(1) Lagrange Polynomials
(2) Polynomial interpolations; Splines

2. Numerical Differentiation (4 hrs)
(1) Construction of finite difference scheme, order of accuracy
(2) Modified wavenumber as a measure of accuracy
(3) Pade approximation
(4) Matrix representation of finite difference schemes

3. Numerical Integration (8 hrs)
(1) Trapezoidal rule; Simpson’s rule; error analysis and mid-point rule
(2) Romberg integration and Richardson’s extrapolation
(3) Adaptive quadrature; Gauss quadrature

4. Numerical Solution of Ordinary Differential Equations (10 hrs)
(1) Initial value problems; numerical stability analysis, model equation
(2) Accuracy; phase and amplitude errors
(3) Runge-Kutta type formulas, multi-step methods; implicit methods
(4) System of differential equations; stiffness
(5) Linearization for implicit solution of non-linear differential equations
(6) Boundary value problems, shooting, direct methods, non-uniform grids, eigenvalue problems

5. Partial Differential Equations (10 hrs)
(1) Finite-difference solution of partial differential equations
(2) Modified wavenumber and Von Neumann stability analysis, modified equations analysis
(3) Alternating direction implicit methods; non-linear equations; iterative methods for elliptic PDE’s