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ENO (Essentially Non-Oscillatory) and WENO (Weighted EssentiallyNon-Oscillatory) schemes are widely used high-order schemes for solving partialdifferential equations (PDEs), especially hyperbolic conservation laws withpiecewise smooth solutions. For structured meshes, these techniques can achievehigh order accuracy for smooth functions while being non-oscillatory neardiscontinuities. For unstructured meshes, which are needed for complexgeometries, similar schemes are required but they are much more challenging. Wepropose a new family of non-oscillatory schemes, called WLS-ENO, in the contextof solving hyperbolic conservation laws using finite-volume methods overunstructured meshes. WLS-ENO is derived based on Taylor series expansion andsolved using a weighted least squares formulation. Unlike other non-oscillatoryschemes, the WLS-ENO does not require constructing sub-stencils, and hence itprovides more flexible framework and is less sensitive to mesh quality. Wepresent rigorous analysis of the accuracy and stability of WLS-ENO, and presentnumerical results in 1-D, 2-D, and 3-D for a number of benchmark problems, andalso report some comparisons against WENO.

WLS-ENO: Weighted-Least-Squares Based Essentially Non-Oscillatory Schemes for Finite Volume Methods on Unstructured Meshes