The multidimensional upwinding method (MDUM) is used for the numerical calculation of concentration, potential and current density distributions at a rotating disc electrode controlled by diffusion, convection, migration and homogeneous reactions of multiple ions. MDUM, previously applied for two-dimensional applications, is now extended to axisymmetrical problems. The electrolyte solutions are supposed to be dilute, at steady state and at a constant temperature. Numerical calculations are performed for two electrochemical systems. In the first system, silver is reduced from a nitrate + thiosulphate solution. This system is controlled by diffusion, convection and a chemical-electrochemical reaction, while migration has a minor influence. In the second system, copper is reduced from a copper sulphate + sulphuric acid solution. Since the amount of sulphuric acid is not excessive, migration has a significant influence on the current density distribution at the electrode. For the copper system, numerical data obtained from MDUM are compared with both experimental results and numerical data from the literature and are found to be in very good agreement. MDUM provides reliable results for both systems without making any simplifications of the governing transport equations for dilute solutions and with a reasonable computational effort.