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This paper reports on the historical development of the Runge-Kutta methods beginning with the simple Euler method up to an embedded 13-stage method. Moreover, the design and the use of those methods under error order, stability and computation time conditions is edited for students of numerical analysis at undergraduate level. The second part presents applications in natural sciences, compares different methods and illustrates some of the difficulties of numerical solutions.
First, we review the current status of the detection of strong `external' variability in the CLASS gravitational B1600+434, focusing on the 1998 VLA 8.5-GHz and 1998/9 WSRT multi-frequency observations. We show that this data can best be explained in terms of radio-microlensing. We then proceed to show some preliminary results from our new multi-frequency VLA monitoring program, in particular the detection of a strong feature (~30%) in the light curve of the lensed image which passes predominantly through the dark-matter halo of the lens galaxy. We tentatively interpret this event, which lasted for several weeks, as a radio-microlensing caustic crossing, i.e. the superluminal motion of a micro-arcsec-scale jet-component in the lensed source over a single caustic in the magnification pattern, that has been created by massive compact objects along the line-of-sight to the lensed image.