Transient simulations of nuclear systems face the computational challenge of resolving both space and time during reactivity changes. A common strategy for tackling this issue is to split the neutron flux into shape and amplitude functions. This split can be solved with high-order/low-order methods. In this paper, a direct comparison of commonly used approximations (e.g., adiabatic, omega, alpha eigenvalue, frequency transform, quasi-static) is performed on the two-dimensional Laboratorium für Reaktorregelung und Anlagensicherung (2D-LRA) benchmark problem using a diffusion solver as the high-order solver and point kinetics as the low-order solver. Additionally, a novel hybrid omega/alpha-eigenvalue solver that incorporates frequencies to model delayed neutrons is introduced. The goal of the comparison is to quantify the performance of each method on a common problem to help inform promising pathways for costly high-fidelity solvers. Overall, we show that exponential frequency approximations are an effective strategy for increasing the accuracy of transient simulations with no added cost. Root-mean-square error of the power distribution at the peak of the transient was consistently decreased by 20% by including frequencies. In particular, the hybrid omega/alpha-eigenvalue method shows improvement over existing eigenvalue solvers as a high-order method. However, in our implementation, the cost of solving for the alpha eigenmode is too costly to recommend over the omega method. While time-differencing schemes are more accurate, we believe the eigenvalue methods are more adaptable to further applications in Monte Carlo transients. Furthermore, they required fewer outer time steps, significantly reducing the computational cost.