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As a first approximation beyond linearity, the nonlinear Schrödinger equation (NLSE) reliably describes a broad class of physical systems. Though numerical solutions of this model are well-established, these methods can be computationally complex. In this paper, we showcase a code development approach, demonstrating how computational time can be significantly reduced with readily available graphics processing unit (GPU) hardware and a straightforward code migration using open-source libraries. This process shows how CPU computations with power-law scaling in computation time with grid size can be made linear using GPUs. As a specific case study, we investigate the Gross-Pitaevskii equation, a specific version of the nonlinear Schrödinger model, as it describes in two dimensions a trapped, interacting, two-component Bose-Einstein condensate (BEC) subject to a spatially dependent interspin coupling, resulting in an analog to a spin-Hall system. This computational approach lets us probe high-resolution spatial features - revealing an interaction-dependent phase transition - all in a reasonable amount of time. Our computational approach is particularly relevant for research groups looking to easily accelerate straightforward numerical simulation of physical phenomena.