A numerical study of the Kelvin-Helmholtz instability in compressible magnetohydrodynamics is presented. The three-dimensional simulations consider shear flow in a cylindrical jet configuration, embedded in a uniform magnetic field directed along the jet axis. The growth of linear perturbations at specified poloidal and axial mode numbers demonstrate intricate nonlinear coupling effects. The physical mechanisms leading to induced secondary Kelvin-Helmholtz instabilities at higher mode numbers are identified. The initially weak magnetic field becomes locally dominant in the nonlinear dynamics before and during saturation. Thereby, it controls the jet deformation and eventual breakup. The results are obtained using the Versatile Advection Code [G. Toth, Astrophys. Lett. Commun. 34, 245 (1996)], a software package designed to solve general systems of conservation laws. An independent calculation of the same Kelvin-Helmholtz unstable jet configuration using a three-dimensional pseudospectral code gives important insights into the coupling and excitation events of the various linear mode numbers. (C) 1999 American Institute of Physics. [S1070-664X(99)01005-8].