The Harris-Nesbet variational method has been considered for an accurate calculation of positron collisions with hydrogen-like ions. We applied our newly developed computer code for positron collisions with hydrogen-like ions to a benchmark calculation at energies below the first excitation threshold of the ions. Very accurate results of phase shift have been obtained for the first time for partial-wave scattering from various hydrogen-like ion targets, with L equal up to 6. The accuracy of the results has been double-checked by carrying out calculations using different large coupling schemes. Our results were compared with those calculated by other research groups who had employed different numerical methods for their calculations.