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Context. We report on new simultaneous observations and modeling of the millimeter, near-infrared, and X-ray flare emission of the source Sagittarius A" (SgrA") associated with the super-massive (4 × 106 M☉) black hole at the Galactic center. Aims. We study the applicability of the adiabatic synchrotron source expansion model and study physical processes giving rise to the variable emission of SgrA" from the radio to the X-ray domain. Methods. Our observations were carried out on 18 May 2009 using the NACO adaptive optics (AO) instrument at the European Southern Observatory's Very Large Telescope, the ACIS-I instrument aboard the Chandra X-ray Observatory, the LABOCA bolometer at the Atacama Pathfinder EXperiment (APEX), and the CARMA mm telescope array at Cedar Flat, California. Results. The X-ray flare had an excess 2-8 keV luminosity between 6 and 12×1033 erg s-1. The observations reveal flaring activity in all wavelength bands that can be modeled as the signal from an adiabatically expanding synchrotron self-Compton (SSC) component. Modeling of the light curves shows that the sub-mm follows the NIR emission with a delay of about three-quarters of an hour with an expansion velocity of about vexp ∼ 0.009c. We find source component sizes of around one Schwarzschild radius, flux densities of a few Janskys, and spectral indices α of about +1 (S (v) ∝ v-α). At the start of the flare, the spectra of the two main components peak just short of 1 THz. To statistically explain the observed variability of the (sub-)mm spectrum of SgrA", we use a sample of simultaneous NIR/X-ray flare peaks and model the flares using a synchrotron and SSC mechanism. Conclusions. These parameters suggest that either the adiabatically expanding source components have a bulk motion larger than vexp or the expanding material contributes to a corona or disk, confined to the immediate surroundings of SgrA". For the bulk of the synchrotron and SSC models, we find synchrotron turnover frequencies in the range of 300-400 GHz. For the pure synchrotron models, this results in densities of relativistic particles of the order of 106.5 cm-3 and for the SSC models, the median densities are about one order of magnitude higher. However, to obtain a realistic description of the frequency-dependent variability amplitude of SgrA", models with higher turnover frequencies and even higher densities are required. [ABSTRACT FROM AUTHOR]