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An interface-tracking numerical algorithm for the simulation of magnetohydrodynamic multiphase/free surface flows in the low-magnetic-Reynolds-number approximation of (Samulyak R., Du J., Glimm J., Xu Z., J. Comp. Phys., 2007, 226, 1532) is described. The algorithm has been implemented in multi-physics code FronTier and used for the simulation of MHD processes in liquids and weakly ionized plasmas. In this paper, numerical simulations of a liquid mercury jet entering strong and nonuniform magnetic field and interacting with a powerful proton pulse have been performed and compared with experiments. Such a mercury jet is a prototype of the proposed Muon Collider/Neutrino Factory, a future particle accelerator. Simulations demonstrate the elliptic distortion of the mercury jet as it enters the magnetic solenoid at a small angle to the magnetic axis, jet-surface instabilities (filamentation) induced by the interaction with proton pulses, and the stabilizing effect of the magnetic field.