Aerosols in many natural and industrial processes are affected by coagulation, the merging of aerosol particles. The present work focuses on a semi-implicit size-discrete method to simulate the evolution of the particle size distribution by coagulation. The derivation of the method is retraced and generalized in terms of the size definition and discretization schemes. Based on this, new forms for the partition coefficients, that implement the redistribution of coagulated particles into the discrete size intervals, are proposed. For this, either constant or linear particle size distribution functions in terms of number and volume concentrations are assumed in each size interval. These assumptions also impact the calculation of the mean volume of the particles in each discretization interval. The inclusion of the coagulation kernel, which describes the physical conditions and the collision mechanism, in the partition coefficients is also tested. The modifications are compared against existing methods in the reproduction of analytical solutions for certain coagulation problems. It was found that certain combinations of modifications were able to reduce the numerical error caused by particle size discretization by more than 80% in the tested cases.