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

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Course Curriculum

Introduction, Why and how we need computers Details 47:54
Representing Arrays and functions on computers Details 40:51
Representing functions – Box functions Details 47:45
Representing functions – Polynomials & Hat functions Details 53:34
Hat functions, Quadratic & Cubic representations Details 50:27
Demo – Hat functions, Aliasing Details 50:58
Representing Derivatives – finite differences Details 50:44
Finite differences, Laplace equation Details 49:32
Laplace equation – Jacobi iterations Details 50:12
Laplace equation – Iteration matrices Details 51:18
Laplace equation – convergence rate Details 33:1
Laplace equation – convergence rate Continued Details 30:23
Demo – representation error, Laplace equation Details 50:57
Demo – Laplace equation, SOR Details 50:56
Laplace equation – final, Linear Wave equation Details 51:24
Linear wave equation – Closed form & numerical solution, stability analysis Details 50:46
Generating a stable scheme & Boundary conditions Details 51:34
Modified equation Details 51:11
Effect of higher derivative terms on Wave equation Details 51:34
Artificial dissipation, upwinding, generating schemes Details 51:48
Demo – Modified equation, Wave equation Details 51:6
Demo – Wave equation / Heat Equation Details 50:3
Quasi-linear One-Dimensional. wave equation Details 31:32
Shock speed, stability analysis, Derive Governing equations Details 51:50
One-Dimensional Euler equations – Attempts to decouple Details 51:6
Derive Eigenvectors, Writing Programs Details 52:14
Applying Boundary conditions Details 50:53
Implicit Boundary conditions Details 51:12
Flux Vector Splitting, setup Roe’s averaging Details 51:13
Roe’s averaging Details 51:59
Demo – One Dimensional flow Details 51:34
Accelerating convergence – Preconditioning, dual time stepping Details 52:41
Accelerating convergence, Intro to Multigrid method Details 53:33
Multigrid method Details 53:31
Multigrid method – final, Parallel Computing Details 52:16
Calculus of Variations – Three Lemmas and a Theorem Details 52:37
Calculus of Variations – Application to Laplace Equation Details 50:56
Calculus of Variations -final & Random Walk Details 52:39
Overview and Recap of the course Details 53:47

This course is delivered by NPTEL, is part of IIT Madras online courses.

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