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This paper is devoted to numerical simulation of the cathode attachment zone of constricted high-current vacuum arcs driven by a transverse magnetic field. With help of particle-in-cell and direct simulation Monte Carlo methods, a 1-D model of the attachment zone was developed, and the stationary-cathode-spot mode was investigated. The states of the charged particles and atoms in the Knudsen layer, the ionization layer, and the plasma sheath are determined in detail. The results support the adaptation of simplified cathode attachment models to be able to reduce the computational effort needed for self-consistent numerical simulations of magnetically driven constricted high-current vacuum arcs.