● **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**

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

● Miscellaneous

- http://nptel.ac.in/syllabus/syllabus.php?subjectId=103106074
- http://www.powershow.com/view/3c158e-ZmE5Y/ME_412_Numerical_Methods_in_Thermal_Science_powerpoint_ppt_presentation
- 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)