The PBE (population balance equation) containing birth, growth, agglomeration and breakage kinetics is described by a conservation law with a moving source term. For the solution of the PBE, we compare two accurate front tracking methods such as a modified method of characteristics (MOC) and a finite difference method with the weighted essentially non-oscillatory (WEND) scheme. Both methods are applied to a potassium sulfate crystallization problem (K2SO4-H2O system) with a discontinuous initial condition. Parameters of agglomeration and breakage kinetics are estimated on the basis of the experimental data of the K2SO4-H2O System. Owing to moving axis along a crystal growth rate (i.e. elimination of the growth term), the modified MOC is able to provide a highly accurate solution even at discontinuous points without numerical diffusion error. However, in the case of stiff nucleation which can commonly appear in practical crystallization processes, it is necessary to adaptively determine time levels to add a mesh of the nuclei size. For solving PBEs involving agglomeration and breakage terms, the MOC can take more long computational time than the spatial discretization methods like the WEND scheme. It is pointed out that the MOC is not available to solve more than two coupled PBEs in general. WENO schemes for spatial discretization are firstly addressed in this study for the dynamic simulation of batch crystallization processes. The WENO schemes show improvements of accuracy and stability over conventional discretization methods (e.g., backward, central or common upwinding schemes). However the WENO schemes on fixed meshes show, to some extent, the numerical diffusion error near discontinuities or steep moving fronts like other finite difference methods. Hence, they require spatially-adaptive mesh techniques in order to track more accurately the moving fronts. Even though the WENO schemes are less accurate than the MOC, they are of practical use for solving complex PBEs owing to a short computational time and little limitation to use.