Introduction - Representation:Need to represent functions on computers - Introduce box functions - Intro to hat functions - Demo representation of sinx using hat functions: Aliasing, high frequency, low frequency. Representation error as a global error. Derivatives of hat functions, Haar functions - Taylor's series, truncation error, representing derivatives - Derivatives of various orders;Simple Problems:Laplace's equation, discretisation, solution - Demo of solution to Laplace's equation. Properties of solution - maximum principle. Proof of uniqueness. Convergence criterion, Jacobi, Gauss-Seidel - Initial condition change for faster convergence, hiearchy of grids, SOR - System of equations, Solution techniques, explanation of SOR- minimization - Matrices, eigenvalues, eigen functions, fixed point theory, stability analysis - Neumann boundary conditions, testing when solution is not known - Wave equation. Physics, directional derivative. Solutions using characteristics. Solution by guessing - Numerical solution - FTCS. Stability analysis - FTFS, FTBS, upwinding, CFL number, meaning, Application of boundary conditions. Physical conditions, numerical conditions - BTCS - stability analysis - Stability analysis of the one - dimensional and two-dimensional heat equations. Connection to solution to Laplace's equation - Modified equation. Consistency. Convergence. Stability - Effect of adding second order, third order fourth order terms to the closed form solution of the wave equation. Dispersion, dissipation - Demo - dissipation, dispersion - Difference between central difference and backward difference. Addition of artificial dissipation to stabilise FTCS - Other schemes - using Taylor's series - Nonlinear wave equation. Non-smooth solution from smooth initial conditions, derivation of the equation as a conservation law. Jump condition - Rankine-Hugoniot relation, speed of the discontinuity - Finite volume method. Finding the flux - Implicit scheme. Delta form, application of boundary conditions. LUAF.
One-D Flow:Derivation of Governing equations. Explanation of the problem. Tentative application of FTCS - Non conservative form. Not decoupled. A r u, p non-conservative. Is there a systematic way to diagonalise. Relation between the two non-conservative forms - Eigenvalues of A'. Eigen vectors., Modal matrix - Stability analysis. Inferred condition. Upwinding. Addition of artificial viscosity - Application of boundary conditions - Demo - solution to one-dimensional flow - Delta form. Application of boundary conditions. Solution technique - Delta form: LU approximate factorization - Finite Volume method. Finding the flux. Roe's Average - Multigrid:Effect of grid size on convergence - why? Geometry. Data transfer two grid correction - Multi- grid more than two grids, V-cycle, W - cycle., work units - Demo + One d Euler equation;Calculus of Variations:Three lemmas and a theorem - Three lemmas and a theorem - problems, ode - Application to Laplace's equation - Closure
Lecture 1: Introduction, Why and how we need computers