Advanced Computational Techniques

Overview

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ABOUT THE COURSE: There are several basic online courses on numerical methods available, however a course which can provide a foundation for the advanced numerical modeling and scientific computing is infrequent. This course is designed as an introductory course on advanced topics on numerical methods for engineering and science students. It is intended to teach the implementation of numerical methods rather than just provide theoretical foundations of the methods. Scientific computing is an integral part of several disciplines including computational mathematics. The completion of this course will equipped the students in handling advanced computational tools. All the methods will be illustrated by working out several examples. Pre-requisite for this course is the basic knowledge of undergraduate calculus and elementary numerical methods.INTENDED AUDIENCE: UG or PG of any Engineering course, Mathematics,Physics and Postgraduate student of Mathematics/Mechanical/Aerospace/Chemical EngineeringPREREQUISITES: Basic UG course in Mathematics/ Undergraduate Calculus

Syllabus

Week 1:Overview on Elementary Numerical Methods, Hermit Interpolation, Cubic Splines, Numerical quadrature, Gauss Quadrature.Week 2:Least square approximation, Linear system of equations, LU-decomposition, Tri-diagonal system, Iterative Methods, SOR, Conjugate gradient method, Eigen value computation, Power method.Week 3:Initial value problems, Predictor-corrector methods, Stability, Truncation error, Runge-Kutta method, Linear boundary value problem, Finite Difference Method.
Week 4:Non-linear BVP, Iterative methods, Advection-diffusion equations, Implicit Scheme, Crank-Nicolson Scheme, von-Neumann stability analysis, Linear hyperbolic PDE, Upwind scheme, Lax scheme, Non-linear Burgers equations, Iterative scheme.