We investigate numerically a one-dimensional linear kinetic equation derived from a velocity jump process modeling bacterial chemotaxis in presence of a chemical signal, produced by cells, and nutrient present in the environment. The model was used to describe accurately lab experiments, in which bands of bacteria are moving across a micro-channel at constant speed. The problem is very different from classical reaction-diffusion traveling waves, as biased transport is the dominant effect here. Calvez et al. obtained analytic results on the existence of traveling waves in the diffusive limit of the velocity-jump process with temporal sensing. The kinetic part of the system is approximated by the well-balanced scheme with the time-space dependent scattering matrix containing coupling with the parabolic system for the concentrations of the chemoattractants. The method gives an accurate resolution in the velocity field and we compare it with the classical time-splitting approach. Then we study numerically the formation and stability of the traveling waves emerging in the kinetic model. Finally, we consider a discrete velocity field and show some sets of parameters for which there exists a numerical cont-example to the existence and stability of traveling waves.