Different algorithms for transient modelling of capillary electrophoresis have been described in several papers. Programs based on such algorithms were applied to various modes of CE. Surprisingly, simulations of capillary isotachophoresis (cITP) at realistic current densities (>1 kA/m2) were not reported. Using these programs for practical cITP conditions resulted in either severe oscillations, mass-balance violation or unexpected program termination. This paper addresses several numerical paths available for modelling one-dimensional capillary electrophoretic behaviour. Tests for determining the validity of the presented solutions with respect to cITP were mass balance checks, zone boundary thickness and the Kohlrausch regulating function. Six different numerical schemes fulfilled these requirements, yet only few could be used for simulating practical current density situations without causing the aforementioned problems. Attention was paid to space discretization (central difference and quadratic upwind) and time integration (implicit, explicit). A single computer program comprising these strategies was developed. Special features for studying transient state phenomena were visualization of concentrations, velocities, Peclet and Courant numbers, electric field strength, conductivity, pH, buffering capacity and charge excess. All parameters could be displayed in both the space domain (profile) as well as in the time domain (electropherogram).