ODEs: Single step methods, Multistep methods, and Hybrid methods for initial value problems( Stiff and Non-stiff) with consistency, stability, convergence and weak stability of these methods.
Finite difference methods for boundary value problems for second order differential equations
Modules
Topics and Contents
Number
of
Lectures
1. Introduction
Preliminaries
Existence, Uniqueness, and Wellposedness
Stability and Asymptotic Stability
3
2. Single Step Methods
The Euler Method
Convergence of Euler’s Method
Improvement of the error bound
Stability
4
3. Higher order Single Step
Methods
Higher Order Methods
Runge-Kutta Methods
Error bounds for Runge-Kutta methods
Absolute Stability for Runge-Kutta Methods
4
4. Systems of Equations and
Equations of Order Greater
Than One
Systems of Equations
Direct Methods For Higher Order Equations
2
5. Consistency, Stability and
Convergence of General
Single – Step Methods
General Single Step Methods
Convergence of General One-Step Methods
2
6. Implicit Runge-Kutta
Methods
Derivation of Implicit Runge-Kutta methods
Derivation of Implicit Runge-Kutta Methods(Contd.)
2
7. Multistep Methods
Multistep Methods
Multistep Methods (Contd.)
Multistep Methods(Contd.)
The local error of the formulas based on integration
Local Error of Nystrom & Milne-Simpson Methods
Multistep Methods for Special Equations of the Second Order
Special 2nd order equations(Contd.)
7
8. Linear Multistep Methods
Linear Multistep Methods
Linear Multistep Methods (Contd)
Consistency and Zero-Stability of Linear Multistep Methods
Convergence of Linear Multistep Methods
Necessary & Sufficient Conditions for Convergence
Absolute Stability and Relative Stability
General methods for finding intervals of absolute and relative
stability
Some more methods for Absolute & Relative Stability
8
9. Stiff-Initial Value Systems
First order linear systems with constant coefficient
Stiffness and Problem of Stiffness
The problem of implicitness for Stiff systems
Linear multistep methods for Stiff systems
4
10. Finite Difference
Methods for Boundary Value
Problems
Finite Difference Methods
Analysis of Difference System
Analytic Expressionof the Error
Nonlinear second order equations
Special Boundary Value Problems
Special Boundary Value Problems(Contd)
6
Total number of lectures
42
Basic course in Numerical analysis
 Lambert, J.D., Computational methods for initial value problemsÂ
Important: Please enable javascript in your browser and download Adobe Flash player to view this site
Site Maintained by Web Studio, IIT Madras. Contact Webmaster: nptel@iitm.ac.in
