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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.
A function has vanishing mean oscillation (VMO) on R up(n) if its mean oscillation - the local average of its pointwise deviation from its mean value - both is uniformly bounded over all cubes within R up(n) and converges to zero with the volume of the cube. The more restrictive class of functions with vanishing lower oscillation (VLO) arises when the mean value is replaced by the minimum value in this definition. It is shown here that each VMO function is the difference of two functions in VLO.
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.
Aspects of open ocean deep convection variability are explored with a two-box model. In order to place the model in a region of parameter space relevant to the real ocean, it is fitted to observational data from the Labrador Sea. A systematic fit to OWS Bravo data allows us to determine the model parameters and to locate the position of the Labrador Sea on a stability diagram. The model suggests that the Labrador Sea is in a bistable regime where winter convection can be either ?on? or ?off?, with both these possibilities being stable climate states. When shifting the surface buoyancy forcing slightly to warmer or fresher conditions, the only steady solution is one without winter convection. We then introduce short-term variability by adding a noise term to the surface temperature forcing, turning the box model into a stochastic climate model. The surface forcing anomalies generated in this way induce jumps between the two model states. These state transitions occur on the interannual to decadal timescale. Changing the average surface forcing towards more buoyant conditions lowers the frequency of convection. However, convection becomes more frequent with stronger variability in the surface forcing. As part of the natural variability, there is a non-negligible probability for decadal interruptions of convection. The results highlight the role of surface forcing variability for the persistence of convection in the ocean.