Introductory Finite Difference Methods for PDEs Professor D M Causon; Professor C G Mingham Download free books at Professor D M Causon & Professor C G Mingham Introductory Finite Difference Methods for PDEs Download free eBooks at bookboon.com Introductory Finite Difference Methods for PDEs © 2010 Professor D M Causon, Professor C G Mingham & Ventus Publishing ApS ISBN 978-87-7681-642-1 Download free eBooks at bookboon.com Introductory Finite Difference Methods for PDEs Contents Contents Preface 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Introduction Partial Differential Equations Solution to a Partial Differential Equation PDE Models Classiication of PDEs Discrete Notation Checking Results Exercise 10 10 10 11 11 15 15 16 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Fundamentals Taylor’s Theorem Taylor’s Theorem Applied to the Finite Difference Method (FDM) Simple Finite Difference Approximation to a Derivative Example: Simple Finite Difference Approximations to a Derivative Constructing a Finite Difference Toolkit Simple Example of a Finite Difference Scheme Pen and Paper Calculation (very important) Exercise 2a Exercise 2b 17 17 17 18 18 20 24 28 32 33 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges An environment in which your expertise is in high demand Enjoy the supportive working atmosphere within our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing light every day Light is OSRAM Click on the ad to read more Download free eBooks at bookboon.com Introductory Finite Difference Methods for PDEs Contents 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 Elliptic Equations Introduction Finite Difference Method for Laplace’s Equation Setting up the Equations Grid Convergence Direct Solution Method Exercise 3a Iterative Solution Methods Jacobi Iteration Gauss-Seidel Iteration Exercise 3b Successive Over Relaxation (SoR) Method Line SoR Exercise 3c 34 34 34 37 38 38 41 42 43 45 47 47 49 51 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Hyperbolic Equations Introduction 1D Linear Advection Equation Results for the Simple Linear Advection Scheme Scheme Design Multi-Level Scheme Design Exercise 4a Implicit Schemes Exercise 4b 52 52 53 55 60 67 69 70 76 360° thinking Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities Click on the ad to read more Download free eBooks at bookboon.com ... Causon & Professor C G Mingham Introductory Finite Difference Methods for PDEs Download free eBooks at bookboon.com Introductory Finite Difference Methods for PDEs © 2010 Professor D M Causon,... bookboon.com Introductory Finite Difference Methods for PDEs Contents 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 Elliptic Equations Introduction Finite Difference Method for Laplace’s... Simple Finite Difference Approximation to a Derivative Example: Simple Finite Difference Approximations to a Derivative Constructing a Finite Difference Toolkit Simple Example of a Finite Difference