Indeed, there is plenty of software for home computers that can do the calculations, and there are lists that tell where eclipses occurred thousands of years ago and give such circumstances as their beginning and ending times to an accuracy of minutes. However, before utilizing such resources one has to clearly understand the approximations inherent in both the input data and in the algorithms for the calculation. A program can work well over short time spans but not over long ones, because planetary motions are essentially nonlinear and initial errors propagate exponentially. Therefore, for an orbit to be precisely defined over a given span of time, the input data have to be sufficiently accurate and all planetary perturbations have to be properly taken into account. The perturbations, particularly, are responsible for many unforeseen effects, including chaos and other unpredictables.
Archaeologists are basically interested in two consequences of such calculations.
One is the determination of the moment the Moon arrives at a point in its orbit opposite the Sun, when it can pass through the Earth's shadow and be eclipsed. The second is to find the local time an observer's clock would show for the event. The physical interrelated effects that contribute to uncertainty include tidally and nontidally induced variations in the Earth's rotation rate and the gradually increasing distance of the Moon from Earth. With events that happened 4,000 years ago, like the Ur eclipses, these effects can result in prediction errors of up to two hours, not merely minutes, despite using the most accurate data available. My search was constrained by the textual evidence in Tablets 20 and 21 of the Enu-ma Anu Enlil.