Introductory Finite Difference Methods for PDEs

1 PROFESSOR D. M. CAUSON & PROFESSOR C. G. MINGHAM Introductory Finite Difference Methods FOR PDES DOWNLOAD FREE TEXTBOOKS AT NO REGISTRATION NEEDED Download free books at Professor D. M. Causon & Professor C. G. MinghamIntroductory Finite Difference Methods for PDEsDownload free books at Introductory Finite Difference Methods for PDEs 2010 Professor D. M. Causon, Professor C. G. Mingham & Ventus Publishing ApSISBN 978-87-7681-642-1 Download free books at ContentsIntroductory Finite Difference Methods for PDEsContents Preface 91. Introduction Partial Differential Equations Solution to a Partial Differential Equation PDE Models Classification of PDEs Discrete Notation Checking Results Exercise 1 162. 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 33what s missing in this equation?

2 MaeRsK inteRnationaL teChnoLogY & sCienCe PRogRammeYou could be one of our future talentsAre you about to graduate as an engineer or geoscientist? Or have you already graduated?If so, there may be an exciting future for you with Moller - Maersk. click the advertDownload free books at ContentsIntroductory Finite Difference Methods for PDEs3. 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 514. 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 click the advertDownload free books at Introductory Finite Difference Methods for PDEs6 Contents5.

3 Parabolic Equations: the Advection-Diffusion Equation Introduction Pure Diffusion Advection-Diffusion Equation Exercise 5b 836. Extension to Multi-dimensions and Operator Splitting Introduction 2D Scheme Design (unsplit) Operator Splitting (Approximate Factorisation) 927. Systems of Equations Introduction The Shallow Water Equations Solving the Shallow Water Equations Example Scheme to Solve the SWE Exercise 7 111 Appendix A: Definition and Properties of Order Definition of O(h) The Meaning of O(h) Properties of O(h) Explanation of the Properties of O(h) Exercise A 114 Sign up for Vestas Winnovation Challenge now- and win a trip around the worldRead more at click the advertDownload free books at Introductory Finite Difference Methods for PDEs7 Contents Appendix B: Boundary Conditions Introduction Boundary Conditions Specifying Ghost and Boundary Values Common Boundary Conditions Exercise B 121 Appendix C: Consistency, Convergence and Stability Introduction Convergence Consistency and Scheme Order Stability Exercise C 133 Appendix D.

4 Convergence Analysis for Iterative Methods Introduction Jacobi Iteration Gauss-Seidel Iteration SoR Iterative Scheme Theory for Dominant Eigenvalues Rates of Convergence of Iterative Schemes Exercise D 143 Graduates for the battle against diabetesNovo Nordisk is the global market leader in diabetes care, with offices in 81 countries and more than employees worldwide. If you want to do the right thing, we need your drive to carry on. Join us at Base Camp, our Graduate ways to unleash a global career As a Graduate with us, you'll discover the responsibility of changing lives, while meeting high-level challenges and gaining hands-on experience every day. And you get to choose between 10 different Graduate paths. You will begin your ascent at either our Danish headquarters in Copenhagen or with our Swiss affiliate in Z rich.

5 Are you top of your field and a keen learner?If you are a top graduate within economics, market-ing, international business, engineering or similar, you're on the right track. If you are also global-minded and have a passion for learning, we can't wait to hear from you. Visit Base Camp at your talent into the lives of click the advertDownload free books at Introductory Finite Difference Methods for PDEs8 Professor Causon and Professor MinghamDepartment of Computing and Mathematics, Manchester Metropolitan University, UKTo our parents and to MagsDownload free books at Introductory Finite Difference Methods for PDEs9 PrefacePrefaceThe following chapters contain core material supported by pen and paper exercises together with computer-based exercises where appropriate. In addition there are web links to: worked solutions, computer codes, audio-visual presentations, case studies, further reading.

6 Codes are written using Scilab (a Matlab clone, downloadable for free from ) and also Matlab. The emphasis of this book is on the practical: students are encouraged to experiment with different input parameters and investigate outputs in the computer-based exercises. Theory is reduced to a necessary minimum and provided in appendices. Web links are found on the following web page: This book is intended for final year undergraduates who have knowledge of Calculus and Introductory level computer programming. Download free books at Introductory Finite Difference Methods for PDEs10 Introduction1. Introduction This book provides an introduction to the Finite Difference method (FDM) for solving partial differential equations (PDEs). In addition to specific FDM details, general concepts such as stability, boundary conditions, verification, validation and grid independence are presented which are important for anyone wishing to solve PDEs by using other numerical Methods and/or commercial software packages.

7 Material is presented in order of increasing complexity and supplementary theory is included in appendices. Partial Differential Equations The following equation is an example of a PDE: )y,x,t(fU)y,x,t(cU)y,x,t(bU)y,x,t(ayyxt ( ) where, t, x, y are the independent variables (often time and space) a, b, c and f are known functions of the independent variables, U is the dependent variable and is an unknown function of the independent variables. partial derivatives are denoted by subscripts: 22yyxtyUU,xUU,tUU etc. The order of a PDE is the order of its highest derivative. A PDE is linear if U and all its partial derivatives occur to the first power only and there are no products involving more than one of these terms. ( ) is second order and linear. The dimension of a PDE is the number of independent spatial variables it contains. ( ) is 2D if x and y are spatial variables.

8 Solution to a Partial Differential Equation Solving a PDE means finding the unknown function U. An analytical ( exact) solution of a PDE is a function that satisfies the PDE and also satisfies any boundary and/or initial conditions given with the PDE (more about these later). Most PDEs of interest do not have analytical solutions so a numerical procedure must be used to find an approximate solution. The approximation is made at discrete values of the independent variables and the approximation scheme is implemented via a computer program. The FDM replaces all partial derivatives and other terms in the PDE by approximations. After some manipulation, a Finite Difference scheme (FDS) is created from which the approximate solution is obtained. The FDM depends fundamentally on Taylor s beautiful theorem (circa 1712!) which is stated in the next chapter.

9 Download free books at Introductory Finite Difference Methods for PDEs11 PDE Models PDEs describe many of the fundamental natural laws ( conservation of mass) so describe a wide range of physical phenomena. Examples include Laplace s equation for steady state heat conduction, the advection-diffusion equation for pollutant transport, Maxwell s equations for electromagnetic waves, the Navier Stokes equations for fluid flow and many, many more. The authors main interest is in solving PDEs for fluid flow problems and details, including pictures and animations, can be found at: Classification of PDEs Second order linear PDEs can be formally classified into 3 generic types: elliptic, parabolic and hyperbolic. The simplest examples are: a) Elliptic: )y,x(fUUyyxx .This is Poisson s equation or Laplace s equation (when f(x,y) =0) which may be used to model the steady state temperature distribution in a plate or incompressible potential flow.

10 Notice there is no time derivative. b) Parabolic: xxtkUU .This is the 1D diffusion equation and can be used to model the time-dependent temperature distribution along a heated 1D bar. c) Hyperbolic: xx2ttUcU .This is the wave equation and may be used to model a vibrating guitar string or 1D supersonic flow. d)xtcUU .This first order PDE is called the advection equation. Solutions of d) also satisfy c). e) xxxtkUcUU .This is the advection-diffusion equation and may be used to model transport of a pollutant in a river. The coefficients k, c in the above PDEs quantify material properties that relate to the problem being solved k could be the coefficient of thermal conductivity in the case of a heated bar, or 1D diffusion coefficient in the case of pollutant transport; c is a wave speed, usually, in fluid flow, the speed of sound. Download free books at Introductory Finite Difference Methods for PDEs12 Initial and Boundary Conditions PDEs require proper initial conditions (ICs) and boundary conditions (BCs) in order to define what is known as a well-posed problem.