IIT Madras Course , Prof. M. Ramakrishna

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IIT Madras Course , Prof. M. Ramakrishna

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

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

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Introduction to CFD by Prof M. Ramakrishna,Department of Aerospace Engineering,IIT Madras.For more details on NPTEL visit httpnptel.ac.in

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IITMadras delivers the above video lessons under NPTEL program, there are more than 6000+ nptel video lectures by other IITs as well.
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- 1.Introduction, Why and how we need computers
- 2.Representing Arrays and functions on computers
- 3.Representing functions - Box functions
- 4.Representing functions - Polynomials & Hat functions
- 5.Hat functions, Quadratic & Cubic representations
- 6.Demo - Hat functions, Aliasing
- 7.Representing Derivatives - finite differences
- 8.Finite differences, Laplace equation
- 9.Laplace equation - Jacobi iterations
- 10.Laplace equation - Iteration matrices
- 11.Laplace equation - convergence rate
- 12.Laplace equation - convergence rate Continued
- 13.Demo - representation error, Laplace equation
- 14.Demo - Laplace equation, SOR
- 15.Laplace equation - final, Linear Wave equation
- 16.Linear wave equation - Closed form & numerical solution, stability analysis
- 17.Generating a stable scheme & Boundary conditions
- 18.Modified equation
- 19.Effect of higher derivative terms on Wave equation
- 20.Artificial dissipation, upwinding, generating schemes
- 21.Demo - Modified equation, Wave equation
- 22.Demo - Wave equation Heat Equation
- 23.Quasi-linear One-Dimensional. wave equation
- 24.Shock speed, stability analysis, Derive Governing equations
- 25.One-Dimensional Euler equations - Attempts to decouple
- 26.Derive Eigenvectors, Writing Programs
- 27.Applying Boundary conditions
- 28.Implicit Boundary conditions
- 29.Flux Vector Splitting, setup Roe’s averaging
- 30.Roe’s averaging
- 31.Demo - One Dimensional flow
- 32.Accelerating convergence - Preconditioning, dual time stepping
- 33.Accelerating convergence, Intro to Multigrid method
- 34.Multigrid method
- 35.Multigrid method - final, Parallel Computing
- 36.Calculus of Variations - Three Lemmas and a Theorem
- 37.Calculus of Variations - Application to Laplace Equation
- 38.Calculus of Variations -final & Random Walk
- 39.Overview and Recap of the course

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