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A linear solver algorithm used by a large-scale unstructured-grid computational fluid dynamics application is examined for a broad range of familiar and emerging architectures. Efficient implementation of a linear solver is challenging on recent CPUs offering vector architectures. Vector loads and stores are essential to effectively utilize available memory bandwidth on CPUs, and maintaining performance across different CPUs can be difficult in the face of varying vector lengths offered by each. A similar challenge occurs on GPU architectures, where it is essential to have coalesced memory accesses to utilize memory bandwidth effectively. In this work, we demonstrate that restructuring a computation, and possibly data layout, with regard to architecture is essential to achieve optimal performance by establishing a performance benchmark for each target architecture in a low level language such as vector intrinsics or CUDA. In doing so, we demonstrate how a linear solver kernel can be mapped to Intel® Xeon™ and Xeon Phi™, Marvell® ThunderX2®, NEC® SX-Aurora™ TSUBASA Vector Engine, and NVIDIA® and AMD® GPUs. We further demonstrate that the required code restructuring can be achieved in higher level programming environments such as OpenACC, OCCA, and Intel® OneAPI™/SYCL, and that each generally results in optimal performance on the target architecture. Relative performance metrics for all implementations are shown, and subjective ratings for ease of implementation and optimization are suggested.