The concept of electrodiffusion based on the Nernst-Planck equations for ionic fluxes coupled with the Poisson equation expressing relation between gradient of the electric field and the charge density is widely used in many areas of natural sciences and engineering. In contrast to the steady-state solutions of the Nernst-Planck-Poisson (abbreviated as NPP or PNP) equations, little is known about the time-dependent behavior of electrodiffusion systems. We present numerical solutions of an NPP system modeling dynamics in the interior of a membrane channel containing an electrolyte after a potential jump at one side of the membrane (voltage-clamp). The NPP equations were solved using the VLUGR2 solver based on an adaptive-grid finite-difference method with an implicit timestepping. The used approach allows for solving the full set of the NPP equations without approximations such as the electroneutrality or constant-field assumptions. Calculations reveal interesting nonlinear time evolution of the ionic concentrations and potential.