Introduction and Mathematical Preliminaries: Types of partial differential equations, Well posed problems, Properties of linear and quasi-linear equations, Physical character of Subsonic and Supersonic flows, Second-order wave equations, System of first order equations, Weak solutions.
Equations of Parabolic type: Finite difference scheme for heat conduction equation, Crank-Nicolson implicit scheme, Analogy with schemes for ordinary differential equations, Implicit methods, Leap-frog and DuFort-Frankel schemes, Operator notation, The Alternating direction implicit (ADI) method.
Finite Difference, Finite Element and Finite Volume Discretisations: Discretisation of derivatives, Order of accuracy, Consistency, Convergence and Stability, Basics of Finite Element and Finite Volume discretisations.
Equations of Hyperbolic type: Explicit schemes, Lax-Wendroff scheme and variants, Implicit schemes, CFL condition, Upwind schemes, Scalar Conservation law, Hyperbolic system of Conservation laws, Second order wave equation, Method of characteristics for second order hyperbolic equations, flux-vector splitting method, Model Convection-Diffusion Equation.
Equations of Elliptic type: The Laplace equation in two dimensions, Iterative methods for system of linear algebraic systems, Solution of the Penta diagonal system, Approximate factorization schemes, Body-fitted and non-body fitted grid generation examples. Multigrid methods, Applications to practical problems of interest.
Equations of Mixed Elliptic-Hyperbolic Type: Tricomi equation, Transonic computations based on TSP model.
Application to Fluid Mechanics: Basic concepts of Fluid mechanics (Fluid kinematics, Lagrangian and Eulerian Specifications, Streamlines, Path lines and Streaklines, Vorticity and Circulation, Streamfunction), Basic equations of Fluid Dynamics, Navier-Stokes equations, Introduction to numerical solutions of Navier-Stokes equations.