# 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

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

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

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