Explicit Runge-Kutta methods and implicit multistep methods utilizing a Newton-Krylov (NK) nonlinear solver are evaluated for a range of configurations of the shallow water dynamical core of the spectral-element Community Atmosphere Model (CAM-SE) to evaluate their computational performance. These configurations are designed to explore the attributes of each method under different but relevant model usage scenarios including varied spectral order within an element, static regional refinement, and scaling to the largest problem sizes. This analysis is performed within the shallow water dynamical core option of a full climate model code base to enable a wealth of simulations for study, with the aim of informing solver development within the more complete hydrostatic dynamical core used for climate research. The limitations and benefits to using explicit RK versus an implicit multistep method with an NK solver, with different parameters and settings, are discussed in light of the trade-offs with MPI communication and memory and their inherent efficiency bottlenecks. Given the performance behavior across the configurations analyzed here, the recommendation for future work using the implicit solvers is conditional based on scale separation and the stiffness of the problem. For the regionally refined configurations, the implicit method has about the same efficiency as the explicit method, without considering efficiency gains from a preconditioner. The potential for improvement using a preconditioner is greatest for higher spectral order configurations, where more work is shifted to the linear solver. Initial simulations with OpenACC directives to utilize a GPU when performing function evaluations show improvements locally, and that overall gains are possible with adjustments to data exchanges.