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