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