Environmental Protection Agency and the National Oceanic and Atmospheric Administration, and taught at Texas A&M University, the University of Colorado, and Imperial College London. Before joining Tufts, he worked for the U.S. Chapra received engineering degrees from Manhattan College and the University of Michigan. His other books include Surface Water-Quality Modeling, Numerical Methods for Engineers, and Applied Numerical Methods with Python.ĭr. This unfortunate situation existed because so much time and drudgery were required to obtain numerical answers using precomputer techniques.Steve Chapra is the Emeritus Professor and Emeritus Berger Chair in the Civil and Environmental Engineering Department at Tufts University. During the precomputer era, signifi cant amounts of energy were expended on the solution technique itself, rather than on problem defi nition and interpretation (Fig. Furthermore, consistent results are elusive because of simple blunders that arise when numerous manual tasks are performed. Manual calculations are slow and tedious. Although in theory such approaches should be perfectly adequate for solving complex problems, in actuality several diffi culties are encountered. ![]() Calculators and slide rules were used to implement numerical methods manually. Finally, graphical techniques are often limited to problems that can be described using three or fewer dimensions. Furthermore, graphical solutions (without the aid of computers) are extremely tedious and awkward to implement. Although graphical techniques can often be used to solve complex problems, the results are not very precise. These graphical solutions usually took the form of plots or nomographs. Graphical solutions were used to characterize the behavior of systems. Consequently, analytical solutions are of limited practical value because most real problems are nonlinear and involve complex shapes and processes. These include those that can be approximated with linear models and those that have simple geometry and low dimensionality. ![]() However, analytical solutions can be derived for only a limited class of problems. These solutions were often useful and provided excellent insight into the behavior of some systems. Solutions were derived for some problems using analytical, or exact, methods. In the precomputer era there were generally three different ways in which engineers approached problem solving: 1. ![]() ![]() PT1.1.1 Noncomputer Methods Beyond providing increased computational fi repower, the widespread availability of computers (especially personal computers) and their partnership with numerical methods has had a signifi cant infl uence on the actual engineering problem-solving process. It is little wonder that with the development of fast, effi cient digital computers, the role of numerical methods in engineering problem solving has increased dramatically in recent years. Although there are many kinds of numerical methods, they have one common characteristic: they invariably involve large numbers of tedious arithmetic calculations. Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations.
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