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Many-core hardware is targeted specifically at obtaining high performance, but reaching high performance is often challenging because hardware-specific details have to be taken into account. Although there are many programming systems that try to alleviate many-core programming, some providing a high-level language, others providing a low-level language for control, none of these systems have a clear and systematic methodology as a foundation. In this article, we propose stepwise-refinement for performance: a novel, clear, and structured methodology for obtaining high performance on many-cores. We present a system that supports this methodology, offers multiple levels of abstraction to provide programmers a trade-off between high-level and low-level programming, and provides programmers detailed performance feedback. We evaluate our methodology with several widely varying compute kernels on two different many-core architectures: a Graphical Processing Unit (GPU) and the Xeon Phi. We show that our methodology gives insight in the performance, and that in almost all cases, we gain a substantial performance improvement using our methodology.