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Representation of mathematical ideas on the computer: numbers, functions, derivative, differential equations.

Simple problems: Solution to Laplace's equation, one-dimensional first order wave equation, heat equation, Finite difference schemes - stability and consistency,  dissipation dispersion, finite volume method.

One-dimensional Euler's equation: Discretisation,  Delta form, application of boundary conditions.

Advanced topics: Roe's averaging, Multigrid Methods, SOR and variational techniques.

 S.No Topic Title Topic Details 1 Introduction Overview of the course. 2 Representation - I Need to represent functions on computers. 3 Representation - II Introduce box functions. 4 Representation - III Intro to hat functions. 5 Representation - IV Demo representation of sinx using hat functions: Aliasing, high frequency, low frequency. Representation error as a global error. Derivatives of hat functions, Haar functions. 6 Representation - V Taylor's series, truncation error, representing derivatives. 7 Representation - VI Derivatives of various orders. 8 Simple Problems - I Laplace's equation, discretisation, solution. 9 Simple Problems - II Demo of solution to Laplace's equation. Properties of solution - maximum principle. Proof of uniqueness. Convergence criterion, Jacobi, Gauss-Seidel. 10 Simple Problems - III Initial condition change for faster convergence, hiearchy of grids, SOR. 11 Simple Problems - IV System of equations, Solution techniques, explanation of SOR- minimization. 12 Simple Problems - V Matrices, eigenvalues, eigen functions, fixed point theory, stability analysis. 13 Simple Problems - VI Neumann boundary conditions, testing when solution is not known. 14 Simple Problems - VII Wave equation. Physics, directional derivative. Solutions using characteristics. Solution by guessing. 15 Simple Problems - VIII Numerical solution - FTCS. Stability analysis. 16 Simple Problems - IX FTFS, FTBS, upwinding, CFL number, meaning, Application of boundary conditions. Physical conditions, numerical conditions. 17 Simple Problems - X BTCS - stability analysis. 18 Simple Problems - XI Stability analysis of the one - dimensional and two-dimensional heat equations. Connection to solution to Laplace's equation. 19 Simple Problems - XII Modified equation. Consistency. Convergence. Stability. 20 Simple Problems - XIII Effect of adding second order, third order fourth order terms to the closed form solution of the wave equation. Dispersion, dissipation. 21 Simple Problems - XIV Demo - dissipation, dispersion. 22 Simple Problems - XV Difference between central difference and backward difference. Addition of artificial dissipation to stabilise FTCS. 23 Simple Problems - XVI Other schemes - using Taylor's series. 24 Simple Problems - XVI 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. 25 Simple Problems - XVII Finite volume method. Finding the flux. 26 Simple Problems - XVIII Implicit scheme. Delta form, application of boundary conditions. LUAF. 27 One-D Flow I Derivation of Governing equations. Explanation of the problem. Tentative application of FTCS. 28 One-D Flow II 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. 29 One-D Flow III Eigenvalues of A'. Eigen vectors., Modal matrix. 30 One-D Flow IV Stability analysis. Inferred condition. Upwinding. Addition of artificial viscosity. 31 One-D Flow V Application of boundary conditions. 32 One-D Flow VI Demo - solution to one-dimensional flow. 33 One-D Flow VII Delta form. Application of boundary conditions. Solution technique. 34 One-D Flow VIII Delta form: LU approximate factorization. 35 One-D Flow IX Finite Volume method. Finding the flux. Roe's Average. 36 Multigrid - 1 Effect of grid size on convergence - why? Geometry. Data transfer two grid correction. 37 Multigrid - II Multi- grid more than two grids, V-cycle, W - cycle., work units. 38 Multigrid - III Demo + One – d Euler equation. 39 Calculus of Variations - I Three lemmas and a theorem. 40 Calculus of Variations - II Three lemmas and a theorem - problems, ode. 41 Calculus of Variations - III Application to Laplace's equation. 42 Closure Recap course.
1. Calculus, Matrix Algebra, Computer Programming and Fluid Mechanics.

1. Elements of CFD. M. Ramakrishna.