Computational Methods in Physics and Engineering

The following sections are included:

• Contents

• Preface to the First Edition

• Preface to the Second Edition

Modern electronic computers owe their origin, to a large extent, to the needs in science and engineering. In the 50 years or so since their appearance, computers have out-performed their original goals of solving numerical problems and keeping tracks of information. They are now an essential tool in almost every aspect of the daily routines of engineers and scientists, from data collection to writing technical reports. The high speed of computation available to us these days opens up not only new ways of carrying out traditional tasks but also new areas of endeavor that have implications going well beyond what we can realize at the moment. Our concern here is limited to a small, albeit important, corner of the role of modern computers in science and engineering, namely some of the general techniques to solve common problems encountered in physics and engineering.

Numerical integration has been an active field in mathematics even before the introduction of computers. This comes from the simple fact that not all integrations can be carried out analytically, and numerical methods, however tedious without the help of a computer, become the only way to solve the problem. Furthermore, the results of many common integrals, such as those for the error function erf(x) and gamma function Γ(x), do not exist as analytical functions and their values must be found numerically. This is quite different from differentiation. For most of the functions encountered in physics and engineering, differentiations can usually be carried out by analytical means. However, there are exceptions. On many occasions, we encounter functions that are given numerically. In this case, the functional form is not available and the derivatives can only be obtained using numerical methods. Furthermore, numerical differentiation is a good way to be introduced to the wide world of finite difference methods for solving a large variety of problems, including differential equations.

A function is a mathematical expression that describes the relation between variables. For simplicity, let us consider a single-valued one with only one independent variable x. If y is a function of x, we can associate a value of y for every one of x. Often the dependence of y on x is given by some known analytical form but this may not always be the case. For example, we can measure the distances d₀, d₁,…, d_N traversed by a car from some fixed reference point at times t₀, t₁, …, t_N, respectively. In this case, the relation between d and the independent variable t is given by N pairs of numbers (t₀, d₀), (t₁, d₁), …, (t_N, d_N). Unless we are provided with some additional information, the relation between distance d and time t is not available to us in terms of an analytical function. What happens if we wish to find out the value of d at some time t that is not one of those measured? If t is within the range of time [t₀, t_N] where measurements have been taken, the value of d may be approximated using interpolation techniques. If t is outside the range, the corresponding technique is called extrapolation. As in any other approximation schemes, it is important that the method we apply to do calculations also supply us with an estimate of the error associated with the result. Interpolation and extrapolation techniques are also useful in cases where the functional relations are known but are too time consuming to evaluate. We shall see several examples of this kind.

A large number of mathematical functions have been developed over the years because of the needs in physics and engineering. In this chapter, we shall examine some of these functions, partly for their intrinsic usefulness and partly as the vehicle to introduce certain computational techniques. Although each function is discussed in relation to a particular type of problem, the interest is usually much broader. However, we shall not try to make any attempt to cover the full range of possible applications for any of the special functions.

With computers taking over many of the tedious tasks, matrix methods are becoming increasingly more popular in solving problems of interest. In this chapter, we shall be mainly concerned with some of the basic numerical calculations involving matrices. Later, we shall apply some of the techniques described here to solve other problems, such as differential equations.

A large part of the activity in science and engineering centers around taking measurements and comparing the results with theoretical expectations. This is a relatively easy task if the data are precise and the theories are well understood. However, such ideal situations occur only rarely. For most of the time, we are more likely to be dealing with cases where physical phenomena are not clear and measurements are imperfect. Under such circumstances, it may be necessary to make conjectures or models of the situation. To compensate for our incomplete knowledge, the models may contain a number of parameters that must be determined empirically. The uncertainties in the data, caused either by the limitations of the instruments used or by the nature of the physical quantities themselves, may contrive to make it difficult to determine these parameters from the measured values. If the number of independent pieces of data is larger than that of free parameters, it is possible to make progress with the help of statistical analyses. Among those commonly used, the method of least squares is by far the most important.

Monte Carlo techniques are useful in solving a variety of computational problems. Earlier, we made use of such an approach in §2-5 for an integral by evaluating the integrand at a randomly selected sample of points. Many problems, from simulation of experimental situations to complicated theoretical computations, can take advantage of such a sampling approach. We shall begin this chapter with a discussion on generating random numbers, the starting point of any Monte Carlo calculations.

Solving differential equations by numerical methods is one of the well developed areas of computational techniques. This arises in part from the needs in a broad range of fields, from engineering to weather forecasting. Besides the method of finite difference — our primary concern in this chapter — finite element methods are also used extensively and they are introduced in the next chapter. The subject of numerical solution to differential equations is vast and we can give here only a brief account of some of the more common approaches used in problems of interest. There are large numbers of excellent specialized books on both numerical and analytical solutions to various types of differential equations and we shall make reference to them at the appropriate places. Software packages and subroutine libraries are also available to handle certain standard forms of differential equations. Although we shall not make any direct reference to them in our introduction to the subject, they are useful in practical applications. There are also advanced finite difference techniques, such as multigrid, that are more appropriate for specialized treatment and we shall not touch upon them here.

In boundary value problems, finite element methods (FEM) are often used instead of the finite difference methods of the previous chapter. Here, the space is divided into a number of small elements, hence the name finite element method. Within each element, the solution is approximated by simple functions, characterized by a few parameters. There are several obvious advantages to such an approach. In the first place, it is easy to handle cases with odd geometrical shapes and such problems occur quite frequently in engineering. Secondly, we have far more flexibility in adjusting the size of individual elements. As a result, it is relatively easy to have finer subdivisions of the space in regions where accuracies are difficult to achieve. As we have seen earlier, this is not easy to do in finite difference methods for partial differential equations.

Finite element analysis has a long tradition and there are large numbers of excellent treatises on the subject, especially in engineering libraries. In this chapter, we shall give an introduction to the subject through a few examples so that the reader can gain a feeling on the subject and, perhaps, even attempt to solve some simple problems. For more details, see, for example, Burnett, [8] Silvester and Ferrari, [50] and Zienkiewicz and Taylor.[60]

The following sections are included:

• Appendix A
  • Decomposition into prime numbers
  • Bit-reversed order
  • Gaussian elimination of a tridiagonal matrix
  • Random bit generator
  • Reduction of higher-order ODE to first-order

• Appendix B List of Fortran Program Examples

• Bibliography

• Index