Computational Mathematics

Models, Methods, and Analysis with MATLAB and MPI, Second Edition

Robert E. White

• Verlag: Chapman and Hall/CRC
• Jahr: 2015
• ISBN:
  9781482235180
• Auflage: 2
• Thema: Mathematik
• Thema: Zahlentheorie

Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI is a unique book covering the concepts and techniques at the core of computational science. The author delivers a hands-on introduction to nonlinear, 2D, and 3D models; nonrectangular domains; systems of partial differential equations; and large algebraic problems requirin

Chapter 1

Discrete Time–Space Models

The first three sections introduce diffusion of heat in one direction. This is an example of model evolution with the simplest model being for the temperature of a well-stirred liquid where the temperature does not vary with space. The model is then enhanced by allowing the mass to have different temperatures in different locations. Because heat flows from hot to cold regions, the subsequent model will be more complicated. In Section 1.4 a similar model is considered, and the application will be to the prediction of the pollutant concentration in a stream resulting from a source of pollution upstream. Both of these models are discrete versions of the continuous model that are partial differential equations. Section 1.5 indicates how these models can be extended to heat and mass transfer in two directions, which is discussed in more detail in Chapters 3 and 4. In the last section variations of the mean value theorem are used to estimate the errors made by replacing the continuous model by a discrete model. Additional introductory materials can be found in G. D. Smith [29], and in R. L. Burden and J. D. Faires [7].

1.1 Newton Cooling Models

1.1.1 Introduction

Many quantities change as time progresses such as money in a savings account or the temperature of a refreshing drink or any cooling mass. Here we will be interested in making predictions about such changing quantities. A simple mathematical model has the form u⁺ = au + b where a and b are given real numbers, u is the present amount and u⁺ is the next amount. This calculation is usually repeated a number of times and is a simple example of an algorithm. A computer is used to do a large number calculations.

Computers use a finite subset of the rational numbers (a ratio of two integers) to approximate any real number. This set of numbers may depend on the computer being used. However, they do have the same general form and are called floating point numbers. Any real number x can be represented by an infinite decimal expansion x = ±(.x₁ · · · x_d · · · )10^e, and by truncating this we can define the chopped floating point numbers.

Let x be any real number and denote a floating point number by

$\begin{array}{ccc}\hfill \mathit{\text{fl}}\left(x\right)& =& \pm \mathrm{.}{x}_1\cdot \cdot \cdot x_d{10}^e\hfill \\ \hfill & =& \pm (x_1\text{/}10+\cdot \cdot \cdot +x_d\text{/}{10}^d){10}^e\mathrm{.}\hfill \end{array}$

This is a floating point number with base equal to 10 where x₁ is not equal to zero, x_i are integers between 0 and 9, the exponent e is also a bounded integer and d is an integer called the precision of the floating point system. Associated with each real number, x, and its floating point approximate number, fl(x), is the floating point error, fl(x) − x. In general, this error decreases as the precision, d, increases. Each...