ADVANCED NUMERICAL METHOD-I (Optional Paper)

Classification of partial differential equations, Elliptic equation, Parabolic equation, Hyperbolic equation, Solution procedures-Finite differerences, Finite elements, Spectral expansion.
 

Finite Difference procedure in one dimension-Tri-diagonal algorithm, Measurement of numerical error, Loss of significance - Pivoting ,Equations with non-constant coefficients, Gradient boundary conditions, First order approximation, Second order approximation, Fictitious nodes,Richardson extrapolation.
 

Elliptic equations in two dimensions- Iterative methods- A general representation of a sparse matrix, The algorithms, Comparison of Gauss-Jacobi and Gauss-Seidel
Diagonal dominance, Non-rectangular domains.

Parabolic equations in One Dimension-  Richardson's method, Bender-Schmidt method,Dufort-Frankel method, Crank-Nicolson method Numerical consistency, Numerical stability Fourier method , Matrix method , Gradient boundary conditions, Numerical convergence - The Lax equivalence theorem.
 

Parabolic equations in two dimensions- Alternating Direction Implicit,Consistency of ADI , Stability of the ADI algorithm, Hyperbolic equations in one space dimension-Characteristics , The CFL condition , Error analysis of the upwind scheme, Fourier analysis of the upwind scheme, The Lax Wendroff scheme .