THE LIGO-VIRGO SCIENTIFIC COLLABORATION RECENTLY REPORTED THE FIRST TWO DIRECT DETECTIONS OF GRAVITATIONAL WAVE (GW) SIGNALS AND DEMONSTRATED THAT THESE EVENTS (GW150914 AND GW151226) WERE PRODUCED BY THE INSPIRAL AND COALESCENCE OF BINARY BLACK HOLES (BHBHS). THIS BREAKTHROUGH MARKS THE BEGINNING OF THE ERA OF GW ASTRONOMY. GW SIGNALS ARE EXPECTED TO BE GENERATED NOT ONLY BY BHBH BINARIES BUT ALSO BY NEUTRON STAR-NEUTRON STAR (NSNS) BLACK HOLE-NEUTRON STAR (BHNS) AND WHITE DWARF-NEUTRON STAR (WDNS) BINARIES AMONG OTHER POTENTIAL SOURCES. THE INSPIRAL AND COALESCENCE OF THESE COMPACT BINARY STARS POSE SOME OF THE MOST CHALLENGING PROBLEMS IN THEORETICAL ASTROPHYSICS. ONLY RECENTLY HAVE ADVANCES IN NUMERICAL RELATIVITY MADE IT POSSIBLE TO EXPLORE THIS TOPIC IN FULL GENERAL RELATIVITY (GR) AND THESE ADVANCES PROVED CRUCIAL FOR GW150914 AND GW151226 ACHIEVING THE 5-SIGMA CONFIDENCE LEVEL OF DETECTION AND FOR ESTIMATING ACCURATELY THE PARAMETERS (E.G. MASSES) OF THE MERGING BHS AND THE REMNANT KERR BHS. THE MERGERS OF NSNSS AND BHNSS HAVE IMPORTANT CONSEQUENCES NOT ONLY FOR GWS BUT ALSO FOR RESOLVING OTHER ASTROPHYSICAL PUZZLES SUCH AS THE ORIGIN OF SHORT GAMMA-RAY BURSTS (SGRBS). WHILE SIMULATIONS OF THESE SYSTEMS IN FULL GR HAVE BEGUN ONLY THE MOST IDEALIZED TREATMENTS HAVE BEEN PERFORMED TO DATE. MORE DETAILED PHYSICS INCLUDING MAGNETIC FIELDS BLACK HOLE SPIN A REALISTIC HOT NUCLEAR EQUATION OF STATE AND NEUTRINO AND ELECTROMAGNETIC (EM) TRANSPORT MUST ALL BE INCORPORATED. ONLY THEN WILL WE BE ABLE TO IDENTIFY AND PROBE RELIABLY FUTURE SOURCES THAT MAY BE DETECTED SIMULTANEOUSLY IN GWS AND EM RADIATION (THE ESSENCE OF "MULTIMESSENGER ASTRONOMY"). LIKEWISE THE COALESCENCE OF BHBHS IS NOW A SOLVED PROBLEM IN GR BUT ONLY IN VACUUM. SIMULATING SUPERMASSIVE BHBH COALESCENCE IN THE GASEOUS ENVIRONMENTS LIKELY TO BE FOUND IN NEARBY GALAXY CORES OR IN MERGING GALAXIES IS CRUCIAL TO IDENTIFYING A "PRECURSOR" AND/OR "AFTERMATH" EM SIGNAL THAT MIGHT ACCOMPANY THE GWS PRODUCED DURING THE LATE INSPIRAL AND MERGER. THE COALESCENCE OF A WDNS HAS ONLY RECENTLY BEEN TREATED IN GR BUT GR IS NECESSARY TO EXPLORE TIDAL DISRUPTION SCENARIOS IN WHICH THE CAPTURE OF WD DEBRIS BY THE NS MAY LEAD TO CATASTROPHIC COLLAPSE OF THE NS. ALTERNATIVELY THE NS MAY SURVIVE AND THE MERGER MIGHT RESULT IN THE FORMATION OF PULSAR PLANETS FROM THE DEBRIS. THE STABILITY OF ROTATING NSS IN THESE AND OTHER SYSTEMS HAS NOT BEEN FULLY EXPLORED IN GR AND THE FINAL FATE OF UNSTABLE STARS HAS NOT BEEN DETERMINED IN MANY CASES ESPECIALLY IN THE PRESENCE OF MAGNETIC FIELDS AND DIFFERENTIAL ROTATION. THESE SYSTEMS WILL BE PROBED OBSERVATIONALLY IN EM BY CURRENT NASA INSTRUMENTS SUCH AS THE HST CHANDRA NUSTAR INTEGRAL SWIFT AND FERMI AND BY FUTURE NASA SPACE OBSERVATORIES SUCH AS JWST WFIRST NICER AND THE X-RAY SURVEYOR. ALSO IN THE FUTURE NASA-SUPPORTED OBSERVATORIES SUCH AS ELISA WILL DETECT GWS. TREATING ALL OF THESE PHENOMENA THEORETICALLY REQUIRES THE SAME COMPUTATIONAL MACHINERY: A FULLY RELATIVISTIC CODE THAT SIMULTANEOUSLY SOLVES EINSTEIN'S EQUATIONS FOR THE GRAVITATIONAL FIELD MAXWELL'S EQUATIONS FOR THE MAGNETIC FIELD THE EQUATIONS OF RELATIVISTIC MAGNETOHYDRODYNAMICS FOR THE PLASMA AND THE EQUATIONS OF RADIATION TRANSPORT FOR THE PHOTONS AND/OR NEUTRINOS ALL IN THREE SPATIAL DIMENSIONS PLUS TIME. RECENT ADVANCES WE HAVE MADE IN CONSTRUCTING SUCH A CODE (THE ILLINOIS GRMHD CODE WITH MOVING-BOX ADAPTIVE MESH REFINEMENT) NOW MAKE IT POSSIBLE FOR US TO SOLVE THESE FUNDAMENTAL CLOSELY RELATED COMPUTATIONAL PROBLEMS SOME FOR THE VERY FIRST TIME.