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Novel methods in recent years have been developed for numerically solving the hydrodynamical and MHD equations relevant to all kinds of astrophysical flows. I will first (briefly) present one such computational technique, where the numerical grid follows the MHD flow using a "moving mesh". I will then present two very distinct astrophysical scenarios to which I have applied this method, specifically gap opening in protoplanetary disks and electromagnetic counterparts to neutron star mergers