Numerical methods, as shall be described in this book, are merely tools used to enable one to replace differential equations governing the subsurface processes with approximation sets of algebraic equations or matrix equations, which are subsequently solved using the methods of linear algebra and requiring the manipulation of computers (Fig. 2.1). If the differential equations were solved exactly by analytic procedures, the solution would appear as some combination of mathematical functions. Subsequent interest in the value of the solution at various locations within a domain of interest would require that the functions be evaluated. Often, when the functions are of a complex form, the computer must be used to determine the values of the function at the points of interest. In many cases the analytical solution will be in terms of an infinite series or some transcendental functions that can be evaluated only approximately. Nevertheless, it is often possible to control the accuracy of the evaluation by careful use of the computer. The steps outlined above do require some facility with number manipulation on the computer and do yield an approximate value of the solution at points of interest. However, the actual steps involve numerical evaluation of an analytical solution to a differential equation rather than numerical solution to the differential equation. The differences between these concepts is the presence of an exact analytical expression as an intermediate step in the former case and the use of an approximation to the differential equation in the latter case.
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- Numerical Methods Applied to Subsurface Hydrology
Gour-Tsyh (George) Yeh
- Springer US
Systemische Notwendigkeit zur Weiterentwicklung von Hybridnetzen