ME 704: Computational Methods in TFE

Motivation

The study of Thermal & Fluid Engineering involves a lot of complex equations like the Navier-Stokes equations, the analytical solutions to which are rarely available. Hence, numerical techniques of obtaining solutions are called for.

In general there is a need to deal with ordinary differential equations & partial differential equations of many forms.

Various numerical methods of obtaining solutions for these equations are taught and compared in this course with special attention paid to error analysis.

● Overview

CMTFE involves great amount of coding hours to be put in. The numerical techniques taught in lectures are immediately to be coded before a deadline (usually a week’s time is given) using a numerical computing environment such as MATLAB or using C++.

Course starts with mathematical concepts like finding roots of polynomials & finding solution to a matrix equation but later focuses more on different schemes used to solve equations numerically and the pros & cons of each scheme.

Obtaining solutions of equations with Initial Value or Boundary value specified is an integral part of the course.

This course may be a good opportunity to refresh your basic coding skills.

Content

  • Taylor Series, Intermediate Value Theorem.Mean value theorem,Taylor’s Theorem,
  • Errors in Computation.Root Finding, Bisection method, Newton Method,Secant Method, Taylor Series and Newton method for Many Variables, Approximations.
  • Gauss Elimination, Partial Pivoting, Matrix Inverse, Cost of computation,Iterative Methods: Jacobi, Gauss-Seidel , Convergence Criterion.
  • Polynomial Interpolation: Lagrange’s Method, Divided Differences.
  • Splines, Chebyshev Polynomials.
  • Least Square Polynomials, Legendre polynomials, Least Square Data fitting,Numerical Integration.
  • Numerical Differentiation.
  • Ordinary Differential Equations, Euler Method.
  • Error analysis in Euler Method, Richardson extrapolation, Stability of Euler Methods
  • Backward Euler and Trapezoidal Methods, Stability, Convergence of iterative methods, Lower & Higher Order Runge-Kutta Methods, Adams-Bashforth and Adams-Moulton Methods.
  • Higher order ODEs, Two point Boundary Value Problems, Parabolic PDEs.
  • Implicit and Crank-Nicholson Schemes for Parabolic PDEs.
  • Consistency of solution methods, Handling non-linearities,
  • Stability of numerical methods for PDEs.
  • Discrete Perturbation Stability Analysis
  • Von Neumann Stability Analysis
  • Errors due to truncation. Elliptic Equations.
  • Hyperbolic Equations: One-step & multi-step methods(Richtmeyer/Lax-Wendroff & McCormack).
  • Application of various single-step methods to a linear problem.
  • Application of single and multi-step methods to a non-linear problem.
  • damping/oscillatory terms and artificial damping.
  • Flux Corrected Transport.

Online Resources

  1. http://www.me.iitb.ac.in/~kiyer/me704/
  2. http://nptel.ac.in/video.php?subjectId=122102009

Offline Resources

  1. Kendall Atkinson, Weimin Han Elementary Numerical Analysis
  2. Numerical-Methods-for-Engineers by Chapra-Canale
  3. Computational fluid dynamics. The basics with applications – JD Anderson
  4. Applied Numerical Methods – Akai

● Miscellaneous

  1. http://nptel.ac.in/syllabus/syllabus.php?subjectId=103106074
  2. http://www.powershow.com/view/3c158e-ZmE5Y/ME_412_Numerical_Methods_in_Thermal_Science_powerpoint_ppt_presentation
  3. https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw

(Written by Ritesh Chaudhari, on the basis of course taken in Autumn 2013-14 under Prof. U.V. Bhandarkar)

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