Syllabus  |   Lectures  |   Downloads  |   FAQ  |   Ask a question  |  
Course Co-ordinated by IIT Madras
Coordinators
 

 

Download Syllabus in PDF format



Untitled Document
 

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.


Important: Please enable javascript in your browser and download Adobe Flash player to view this site
Site Maintained by Web Studio, IIT Madras. Contact Webmaster: nptel@iitm.ac.in