Mathematics
Numerical Methods
Numerical methods involve using mathematical algorithms to solve problems that are difficult or impossible to solve analytically. These methods often rely on approximations and computational techniques to find solutions to equations, integrals, and other mathematical problems. They are widely used in various fields such as engineering, physics, and computer science for practical problem-solving.
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eBook - PDFDesign Engineer's Reference Guide
Mathematics, Mechanics, and Thermodynamics
- Keith L. Richards(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
29 Introduction to Numerical Methods 2.1 INTRODUCTION Before the advent of electronic digital computers, all computing was accomplished using manual methods. The aim of this introduction is to present some of the methods used in analysing practical prob-lems arising in mechanical engineering that are not generally solvable using classical methods. These methods were developed prior to the development of the digital computer and have stood the test of time. The basic mathematics used in numerical analysis is covered in Chapter 1. The analysis of a physical problem will involve four basic steps: 1. Development of a suitable mathematical model that realistically represents the physical system 2. Derivation of the systems governing equations 3. Solution of the governing equations 4. Interpretation of the results While an analytical solution using classical methods will be exact, if it exists, a numerical method will require a number of iterations to generate a solution; this is only an approximation and cannot be considered to be exact by any means. It is important to understand the errors that arise in numerical analysis, and for this reason a sepa-rate section devoted to errors is covered in Section 2.2 and covers the difference between round-off errors and truncation errors. The Numerical Methods will be able to solve most complex problems, and one of the advantages is easy programming on a computer using programs such as MATLAB ® , MathCAD, Maple and Mathematica. All of these programs are user-friendly. 2.2 Numerical Methods FOR INTEGRATION There may be a point where the engineer is required to calculate a definite integral that cannot be solved using analytical integration or it is preferred to integrate tabulated data. The following methods may be used to solve for a definite integral: 1. Manual method 2. Mid-ordinate rule 3. Trapezoidal rule 4. Simpson’s rule 5. Quadratic triangulation 6. Romberg integration 7. Gauss quadratic 2
eBook - PDFComputational Photonics
An Introduction with MATLAB
- Marek S. Wartak(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
B Appendix B Summary of basic Numerical Methods In a book like this, the development of computer programs for various tasks and also execution of simulations for different processes and devices, plays an essential role. The fundamentals of many computer programs are supported by Numerical Methods. Therefore, in this Appendix we summarize main elements of numerical analysis with an emphasis on methods related to the development of programs used in this book, and also to understanding of operation of those programs. There are many excellent textbooks devoted to numerical analysis. We found the books by Koonin [1], DeVries [2], Garcia [3], Gerald and Wheatley [4], Rao [5], Heath [6] and Recktenwald [7] of significant pedagogical value. The books by Press et al. [8] stand on their own as an excellent source of practical computer codes ready to use. We concentrate on description and implementation of some practical Numerical Methods and not on the problems which those methods are typically used for. We start our discussion with a summary of methods of solving nonlinear equations. There are many textbooks aimed to the introduction of Numerical Methods and their ap- plications. Some of the most popular are: Applied Numerical Analysis Using MATLAB by Fausett [9], Numerical Methods for Physics by Garcia [3], Introduction to Scientific Com- puting by van Loan [10], Advanced Engineering Mathematics with MATLAB by Harman et al. [11], A Friendly Introduction to Numerical Analysis by Bradie [12]. They contain extensive code written in MATLAB (sometimes also in other languages) which should help with understanding and could be easily adopted to a particular problem. Our aim here is to summarize some of the Numerical Methods and techniques which are directly relevant to the problems discussed in the main text. In Table B.1 we provide a list of MATLAB files created for this Appendix B and a short description of each function.
eBook - PDF- Weiming Wu(Author)
- 2007(Publication Date)
- CRC Press(Publisher)
Chapter 4 Numerical Methods River engineering problems are usually governed by nonlinear differential equations in irregular and movable domains, most of which have to be solved using Numerical Methods. Introduced in this chapter are the discretization methods for 1-D, 2-D, and 3-D problems on fixed and moving grids, the solution strategies for the Navier-Stokes equations, and the solution methods of algebraic equations. Some of these can be found in Patankar (1980), Hirsch (1988), Fletcher (1991), Ferziger and Peric (1995), Shyy et al. (1996), etc. 4.1 CONCEPTS OF NUMERICAL SOLUTION 4.1.1 General procedure of numerical solution Consider the problem in a domain of a ≤ x ≤ b shown in Fig. 4.1, governed by a differential equation L( f ; x) = S (4.1) with boundary conditions f | x=a = f a , f | x=b = f b (4.2) where L is the differential operator, f is the function to be determined, x is the spatial coordinate, and S is the source term. To acquire a numerical solution, the study domain is first represented by a finite number of points, denoted as x 1 , x 2 , . . . , and x N , which constitute the computational grid (mesh). Here, x 1 = a and x N = b. The distance between two consecutive points, x, is the grid size or spacing. Eq. (4.1) is discretized on the computational grid using a numerical method. A discrete equation L d is then established to approximate the differential equation at each grid point: L d ( ˆ f ; x i ) = S i (i = 2, . . . , N − 1) (4.3) 114 Computational River Dynamics Figure 4.1 Example of numerical solution. where ˆ f is the approximate solution of f , which is subject to boundary conditions: ˆ f 1 = f a , ˆ f N = f b (4.4) where ˆ f 1 and ˆ f N are the values of ˆ f at x = x 1 and x N , respectively. The system of algebraic equations consisting of discrete equations (4.3) and bound- ary conditions (4.4) is used to determine the approximate solution ( ˆ f 1 , ˆ f 2 , . . . , ˆ f N ) on the computational grid.
- Kevin W. Cassel(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
The primary distinction, and the one that matters most when selecting appropriate Numerical Methods, is whether the differen- tial equation is an initial-value problem or a boundary-value problem. The mathemat- ical structure of each is fundamentally different, and this influences the development of finite-difference methods for each type of problem. 7.1 General Considerations 377 7.1 General Considerations 7.1.1 Numerical Solution Procedure Before getting into a detailed discussion of Numerical Methods for solving differential equations, it is helpful to discuss the numerical solution procedure in the form of a general framework as illustrated in Figure 7.1. We begin with the physical system of interest to be modeled. The first step is to apply appropriate physical laws and models in order to derive the mathematical model, or governing equations, of the system. These are typically in the form of ordinary or partial differential equations. The physical laws include, for example, conservation of mass, momentum, and energy, and models include any assumptions or idealizations applied in order to simplify the governing equations. When possible, analytical solutions of the mathematical model are sought. If this is not possible, we turn to Numerical Methods, which is the focus of Part II. The second step is to discretize the mathematical model, which involves approximation of the continuous differential equation(s) by a system of algebraic equations for the dependent variables at discrete locations in the independent variables (space and time). In this way, the exact and continuous mathematical model is con- verted into a system of linear algebraic equations that can be solved using the methods outlined in the previous chapter. For example, see Figure 7.2. The discretization step leads to a system of linear algebraic equations, whose numerical solution comprises step 3 of the numerical solution procedure.
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