There is a strong need for high-accuracy and efficient modeling of extreme-mass-ratio binary black hole systems because these are strong sources of gravitational waves that would be detected by future observatories. In this article, we present sample results from our Teukolsky EMRI code: a time-domain Teukolsky equation solver (a linear, hyperbolic, partial differential equation solver using finite-differencing), that takes advantage of several mathematical and computational enhancements to efficiently generate long-duration and high-accuracy EMRI waveforms.We emphasize here the computational advances made in the context of this code. Currently there is considerable interest in making use of many-core processor architectures, such as Nvidia and AMD graphics processing units (GPUs) for scientific computing. Our code uses the Open Computing Language (OpenCL) for taking advantage of the massive parallelism offered by modern GPU architectures. We present the performance of our Teukolsky EMRI code on multiple modern processor architectures and demonstrate the high level of accuracy and performance it is able to achieve. We also present the code's scaling performance on a large supercomputer i.e. NSF's XSEDE resource, Keeneland 1 .