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. |