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High-order finite volume schemes for unstructured meshes first appeared in the literature at least as early as Barth and Frederickson's 1990 paper. 1 However, the approach has not gained a wide following, perhaps because of the difficulties in achieving a genuinely high-order accurate solution, especially when dealing with curved boundaries. In this paper, we document in detail our approach to constructing a high-order solver. In addition to reconstruction, we discuss flux integration and curved boundary treatment. We describe how we tackle each of these issues, provide useful testing procedures to verify code correctness, and illustrate the results of mis-handling various small but important details.

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