A new numerical method is presented for the calculation of concentration, potential and current distributions in two-dimensional electrochemical cells controlled by diffusion, convection and migration of ions. The numerical model, for reasons of generality developed for an explicit time-dependent solution, has been made implicit such that it can easily deal with electrochemical cells at steady-state involving multiple ions. The electrolyte solutions are considered to be dilute and at a constant temperature. This new method, the multi-dimensional upwinding method, originates from the field of fluid mechanics. It is an alternative approach to both finite element methods and finite volume methods. In order to evaluate the method, steady-state computations have been performed on two plane, parallel electrodes embedded in the walls of a flow channel. Tertiary current distributions have been calculated using Butler-Volmer polarisation laws and with the applied cell voltage as a driving force for concentration and potential gradients. Electrolytes with two and three ions were considered and the results, in case of an excessive amount of supporting electrolyte, were compared with the Levecque solution. In all cases, the numerical results are found to be in good agreement with analytical or numerical solutions from literature.