Technische Universit¨atDresden Faculty for Forestry, Geosciences and Hydrosciences Systems Analysis in Water Management Peter-Wolfgang Gr¨aber Summer Semester 2010 This lecture material is only internal to use for the study courses of the Department Hydro Sciences of the TU Dresden. This books are subject to the copyright and is only for the intern use in connection of the education inside of the Technische Universitaet Dresden. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval, without permission in writing form the publisher. Editing and layout: Prof. Dr.-Ing. habil. Peter-Wolfgang Gr¨aber Technische Universit¨atDresden Faculty of Forest, Geosciences and Hydrosciences Institute of Waste Management and Contaminated Site Treatment Phone: +49 (0) 3501 530029 Fax: +49 (0) 3501 530022 e-mail: [email protected] Internet: http://www.tu-dresden.de/fghhiaa Issue: 28.02.2009 II Author presentation The lecture Systems Analysis in Water Management is based on foundation courses such as mathematics, physics and informatics. And the chapters, which play a role in water management task solutions, will be specially discussed in the lecture, such as vector analysis, solution of equation systems, matrix calculation, and solution of differential equations as well as numerical integration. The following chapters are revisions of basic knowledge and will only be shown with corresponding key points. Self study is strongly recommended during the revision. The further chapters go beyond basic knowledge and indicate mathematical methods, which are related to the water management practice. The teaching contents of the subject Systems Analysis in Water Management require an advanced mathematical knowledge, including abstraction ability. In the ex- ercise and computer courses, some problem will be discussed combined with practically relevant cases in order to develop a deeper understanding of this lecture. III Table of contents Author presentation . III List of tables . .XIV List of Figures . .XXI Abbreviations and Formula Symbols . .XXII I Mathematical fundamentals 1 1 Algebra fundamentals . 5 1.1 Exponential and logarithmic expressions . 6 1.1.1 Exercises . 8 1.2 Matrices . 9 1.2.1 Fundamentals . 9 1.2.2 Calculation rules . 14 1.2.3 determinant of a matrix . 19 1.2.4 Exercises . 21 1.3 Linear Equation Systems . 22 1.3.1 Total step method . 24 1.3.1.1 Gauss elimination . 24 1.3.1.2 Cramer's rule . 28 1.3.1.3 Construction of the inverse matrix . 31 IV 1.3.1.4 LU decomposition . 33 1.3.1.5 Cholesky decomposition . 38 1.3.2 Iterative methods . 48 1.3.3 Overdetermined equation systems (m > n) . 49 1.3.4 Exercises . 50 2 Vector algebra and vector calculus . 51 2.1 Unit vectors . 53 2.2 Algorithms . 55 2.3 Examples of vector calculus . 62 2.4 Exercises . 66 3 Interpolation methods . 69 3.1 Polynomial interpolation . 74 3.1.1 Analytical power function . 75 3.1.2 Lagrange interpolation formula . 78 3.1.3 Newton interpolation formula . 81 3.1.3.1 Arbitrary supporting points . 81 3.1.3.2 Equidistant supporting point distribution . 84 3.1.3.3 Examples for the Newton method . 87 3.2 Spline interpolation . 91 3.3 Kriging method . 100 3.4 Exercises . 105 V 4 Optimisation problems. 107 4.1 Analytical solution of extreme value problems . 108 4.2 Iterative optimum search . 108 4.3 Least squares method (MKQ method) . 108 4.4 Search strategies . 109 4.4.1 Jones spiral method . 111 5 Ordinary differential equations. 113 5.1 Setting up equations . 116 5.1.1 Examples of setting up differential equations: . 117 5.1.2 Exercises for setting up ODE's: . 121 5.2 Analytical solution methods . 125 5.2.1 First order ordinary differential equations . 125 5.2.1.1 Solution of homogeneous differential equations . 125 5.2.1.2 Solution of the inhomogeneous first order ODE . 128 5.2.1.3 Exercises . 133 5.2.2 Ordinary differential equations of higher order . 136 5.2.2.1 OED of type a . 136 5.2.2.2 ODE of type b . 137 5.2.2.3 ODE of type c . 142 5.2.2.4 Exercises . 143 5.3 Integral transforms . 144 VI 5.3.1 Time- and Frequency domain . 144 5.3.2 Laplace Transform . 147 5.3.2.1 Forward transformation . 147 5.3.2.2 Important calculation rules . 148 5.3.2.3 Inverse transformation . 150 5.3.2.4 Correspondence table . 151 5.3.3 Solution of differential equations by means of Laplace transform . 153 5.3.3.1 Algorithm . 153 5.3.3.2 Examples . 153 5.3.3.3 Example of ODE systems . 156 5.3.3.4 Exercises to the integral transforms . 158 5.4 Methods for Numerical Integration . 161 5.4.1 Integration . 161 5.4.1.1 Rectangle rule . 161 5.4.1.2 Trapezoidal rule . 164 5.4.1.3 Simpson' rule . 165 5.4.1.4 Newton formula . 165 5.4.1.5 Examples for application of numerical integration . 166 5.4.1.6 Exercises for the application of the numerical integration . 169 5.4.2 Numerical Solutions of Differential Equations . 171 VII 5.4.2.1 Euler method . 172 5.4.2.2 Runge Kutta method . 175 5.4.2.3 Predictor Corrector Method . 178 5.4.2.4 Exercises to numerical solutions of ODE . 181 II Partial differential equations of underground processes 183 6 Overview . 185 6.1 One dimensional flow equation . 188 6.2 Horizontal plane groundwater flow equation . 189 6.3 One dimensional material transfer . 190 6.4 Multiphase flow . 190 7 Horizontal plane Groundwater flow equation . 193 7.1 Dupuit assumption and ballance equation . 194 7.2 Potential illustration . 197 7.3 Boundary conditions . 201 7.3.1 Initial conditions . 201 7.3.2 Randbedingungen . 202 8 Analytical Solutions . 205 8.1 Theis well equation (rotationally symmetrical flow) . 206 8.1.1 General solution . 206 8.1.2 Consideration of special effects . 214 8.1.2.1 Imperfect well . 214 VIII 8.1.2.2 Multi-well plants . 216 8.1.2.3 Variable conveying curve of a well . 217 8.1.2.4 Limitations . 222 8.1.2.5 Multilateral boundary . 228 8.1.3 Supply from neighbouring layers . 231 8.2 Exercises to analytical solutions . 234 9 Numerical methods . 241 9.1 Methods of local quantization . 244 9.1.1 Finite Differencen Method . 246 9.1.1.1 Balance equation . 247 9.1.1.2 Consideration of boundary conditions . 260 9.1.2 Finite Element Method . 266 9.2 Time quantization methods . 270 9.2.1 Backward difference - Implicit methods . 272 9.2.2 Mixed methods . 277 9.2.3 Extrapolation method . ..