COMPUTATIONAL METHODS OF PARTIAL DIFFERENTIAL EQUATIONS (Optional Paper)

Elliptic equations: Finite difference method on 2D and 3D elliptic equation on non-uniform mesh, Finite difference methods for 2D and 3D Poisson’s equations of second and fourth order approximations, Iterative methods for 2D and 3D elliptic equations, Solution of large system of algebraic equations corresponding to discrete problems and iterative methods (Jacobi, Gauss-Seidel and SOR), Numerical methods extended method 2D and 3D bi-harmonic problems.

Heat Equations: Compatibility , Consistency and convergence of the difference method, Numerical methods for one dimensional heat conduction equation: Schmidth scheme, Laasonen scheme,  Cranck Nicholson Scheme, Alternating direction implicit (ADI) methods  for 2D and 3D heat conduction equations,  Stability analysis (Energy method , Matrix method and Von-Neumann method).

First order hyperbolic equation:  Conservation laws, Explicit and implicit methods for diffusion equations, Explicit and implicit difference scheme for first order hyperbolic equations and their stability analysis, System of equation for first order hyperbolic equation, Conservative form, Alternating direction implicit (ADI ) methods for 2D and 3D first order hyperbolic equation.

Second order hyperbolic equations: Methods of characteristic  for evolution problem of hyperbolic type, Von-Neumann method for stability analysis, Explicit and implicit method for second order hyperbolic equation, Operator splitting methods for 2D and 3D wave equations and their stability analysis, Unconditional stability  analysis for second order hyperbolic equations.

Finite element method: Finite element method for second order elliptic BVPS, Finite element equation, Variational problems, Triangular and rectangular finite elements, Standard examples of finite elements, Mixed finite element methods.