Initial Value Problems (IVP) and existence theorem. Truncation error, deriving finite difference equations – Single step methods for I order IVP- Taylor series method, Euler method, Picard’s method of successive approximation – Runge Kutta Methods. Stability of single step methods – Multi step methods for I order IVP – Predictor-Corrector method, Euler PC method, Milne and Adams Moulton PC method – Stability of multi step methods, root condition – System of first order ODE, higher order IVPs – Linear Boundary Value Problems (BVP), finite difference methods, shooting methods, stability, error and convergence analysis – Non linear BVP, higher order BVP – Classification of PDEs, Finite difference approximations to partial derivatives – Solution of one dimensional heat conduction equation by Explicit and Implicit schemes (Schmidt and Crank Nicolson methods ), stability and convergence criteria – Laplace equation using standard five point formula and diagonal five point formula, Iterative methods for solving the linear systems – Hyperbolic equation, explicit / implicit schemes, method of characteristics. Solution of wave equation – Solution of I order Hyperbolic equation. Von Neumann stability

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Course Curriculum

Motivation with few Examples Details 53:54
Single – Step Methods for IVPs Details 55:2
Analysis of Single Step Methods Details 55:51
Runge – Kutta Methods for IVPs Details 56:9
Higher Order Methods/Equations Details 55:38
Error – Stability – Convergence of Single Step Methods Details 58:4
Tutorial – I Details 58:36
Tutorial – II Details 57:24
Multi-Step Methods (Explicit) Details 55:9
Multi-Step Methods (Implicit) Details 53:49
Convergence and Stability of multi step methods Details 59:21
General methods for absolute stability Details 56:45
Stability Analysis of Multi Step Method Details 54:37
Predictor – Corrector Methods Details 57:26
Mod-15 Lec 15 Some Comments on Multi – Step Methods Details 55:24
Finite Difference Methods – Linear BVPs Details 58:24
Mod-17 Lec 17 Linear/Non – Linear Second Order BVPs Details 56:34
BVPS – Derivative Boundary Conditions Details 55:29
Higher Order BVPs Details 58:4
Shooting Method BVPs Details 59:23
Tutorial – III Details 57:45
Introduction to First Order PDE Details 53:3
Introduction to Second Order PDE Details 57:11
Finite Difference Approximations to Parabolic PDEs Details 56:58
Implicit Methods for Parabolic PDEs Details 55:17
Consistency, Stability and Convergence Details 54:48
Other Numerical Methods for Parabolic PDEs Details 57:55
Tutorial – IV Details 55:28
Matrix Stability Analysis of Finite Difference Scheme Details 53:50
Fourier Series Stability Analysis of Finite Difference Scheme Details 53:1
Finite Difference Approximations to Elliptic PDEs- I Details 53:16
Finite Difference Approximations to Elliptic PDEs – II Details 58:9
Finite Difference Approximations to Elliptic PDEs – III Details 57:48
Finite Difference Approximations to Elliptic PDEs – IV Details 57:26
Finite Difference Approximations to Hyperbolic PDEs – I Details 57:46
Finite Difference Approximations to Hyperbolic PDEs – II Details 56:37
Method of characteristics for Hyperbolic PDEs – I Details 55:7
Method of characterisitcs of Hyperbolic PDEs – II Details 59:28
Finite Difference Approximations to 1st order Hyperbolic PDEs Details 54:41
Summary, Appendices, Remarks Details 55:28

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