Numerical Methods in Engineering and Science

This book is designed for an introductory course in numerical methods for students of engineering and science at universities and colleges of advanced education. It is an outgrowth of a course of lectures and tutorials (problem­ solving sessions) which the author has given for a number of years at the University of New South Wales and elsewhere. The course is normally taught at the rate of 1i hours per week throughout an academic year (28 weeks). It has occasionally been given at double this rate over half the year, but it was found that students had insufficient time to absorb the material and experiment with the methods. The material presented here is rather more than has been taught in anyone year, although all of it has been taught at some time. The book is concerned with the application of numerical methods to the solution of equations - algebraic, transcendental and differential - which will be encountered by students during their training and their careers. The theoretical foundation for the methods is not rigorously covered. Engineers and applied scientists (but not, of course, mathematicians) are more con­ cerned with using methods than with proving that they can be used. However, they 'must be satisfied that the methods are fit to be used, and it is hoped that students will perform sufficient numerical experiments to con­ vince themselves of this without the need for more than the minimum of theory which is presented here.

1 Introduction.- 1.1 What are numerical methods?.- 1.2 Numerical methods versus numerical analysis.- 1.3 Why use numerical methods?.- 1.4 Approximate equations and approximate solutions.- 1.5 The use of numerical methods.- 1.6 Errors.- 1.7 Non-dimensional equations.- 1.8 The use of computers.- 2 The solution of equations.- 2.1 Introduction.- 2.2 Location of initial estimates.- 2.3 Interval halving.- 2.4 Simple iteration.- 2.5 Convergence.- 2.6 Aitken’s extrapolation.- 2.7 Damped simple iteration.- 2.8 Newton-Raphson method.- 2.9 Extended Newton’s method.- 2.10 Other iterative methods.- 2.11 Polynomial equations.- 2.12 Bairstow’s method 56 Worked examples 58 Problems.- 3 Simultaneous equations.- 3.1 Introduction.- 3.2 Elimination methods.- 3.3 Gaussian elimination.- 3.4 Extensions to the basic algorithm.- 3.5 Operation count for the basic algorithm.- 3.6 Tridiagonal systems.- 3.7 Extensions to the Thomas algorithm.- 3.8 Iterative methods for linear systems.- 3.9 Matrix inversion.- 3.10 The method of least squares.- 3.11 The method of differential correction.- 3.12 Simple iteration for non-linear systems.- 3.13 Newton’s method for non-linear systems.- Worked examples.- Problems.- 4 Interpolation, differentiation and integration.- 4.1 Introduction.- 4.2 Finite difference operators.- 4.3 Difference tables.- 4.4 Interpolation.- 4.5 Newton’s forward formula.- 4.6 Newton’s backward formula.- 4.7 Stirling’s central difference formula.- 4.8 Numerical differentiation.- 4.9 Truncation errors.- 4.10 Summary of differentiation formulae.- 4.11 Differentiation at non-tabular points: maxima and minima.- 4.12 Numerical integration.- 4.13 Error estimation.- 4.14 Integration using backward differences.- 4.15 Summary of integration formulae.- 4.16 Reducing the truncation error 146 Worked examples 149 Problems.- 5 Ordinary differential equations.- 5.1 Introduction.- 5.2 Euler’s method.- 5.3 Solution using Taylor’s series.- 5.4 The modified Euler method.- 5.5 Predictor-corrector methods.- 5.6 Milne’s method, Adams’ method, and Hamming’s method.- 5.7 Starting procedure for predictor-corrector methods.- 5.8 Estimation of error of predictor-corrector methods.- 5.9 Runge-Kutta methods.- 5.10 Runge-Kutta-Merson method.- 5.11 Application to higher-order equations and to systems.- 5.12 Two-point boundary value problems.- 5.13 Non-linear two-point boundary value problems 198 Worked examples 199 Problems.- 6 Partial differential equations I — elliptic equations.- 6.1 Introduction.- 6.2 The approximation of elliptic equations.- 6.3 Boundary conditions.- 6.4 Non-dimensional equations again.- 6.5 Method of solution.- 6.6 The accuracy of the solution.- 6.7 Use of Richardson’s extrapolation.- 6.8 Other boundary conditions.- 6.9 Relaxation by hand-calculation.- 6.10 Non-rectangular solution regions.- 6.11 Higher-order equations 238 Problems.- 7 Partial differential equations II — parabolic equations.- 7.1 Introduction.- 7.2 The conduction equation.- 7.3 Non-dimensional equations yet again.- 7.4 Notation.- 7.5 An explicit method.- 7.6 Consistency.- 7.7 The Dufort-Frankel method.- 7.8 Convergence.- 7.9 Stability.- 7.10 An unstable finite difference approximation.- 7.11 Richardson’s extrapolation 261 Worked examples 262 Problems.- 8 Integral methods for the solution of boundary value problems.- 8.1 Introduction.- 8.2 Integral methods.- 8.3 Implementation of integral methods 271 Worked examples 278 Problems.- Suggestions for further reading.