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In this thesis, we mainly investigate the application of a nodal discontinuous Galerkin spectral element method (DGSEM) for simulating processes in column liquid chromatography. Additionally, we investigate the effectiveness of a total variation diminishing in the mean (TVDM) limiter in controlling spurious oscillations related to the Gibbs phenomenon. With an order-of-accuracy test, we demonstrated that our nodal DGSEM achieved and, in multiple instances, even exceeded theoretical convergence rates, especially with an increased number of elements, validating the use of high-order basis functions for achieving high-order accuracy. We also demonstrated how setup parameters could affect process outcomes, which suggests that numerical simulations can help guide the development of experimental methods since they can explore the solution space of an optimization problem much faster than experimental procedures by leveraging computational speed. Finally, we showed that the TVDM limiter successfully eliminated severe oscillations and negative concentrations near shock regions but introduced significant smearing of the shocks. These findings validate the nodal DGSEM as a highly accurate and reliable tool for detailed modeling of column liquid chromatography, which is essential for improving efficiency, yield, and product quality in biopharmaceutical manufacturing.