Quasi-ENO Schemes for Unstructured Meshes Based on Unlimited Data-Dependent Least-Squares Reconstruction

A crucial step in obtaining high-order accurate steady-state solutions to the Euler and Navier?Stokes equations is the high-order accurate reconstruction of the solution from cell-averaged values. Only after this reconstruction has been completed can the flux integral around a control volume be accurately assessed. In this work, a new reconstruction scheme is presented that is conservative, is uniformly accurate, allows only asymptotically small overshoots, is easy to implement on arbitrary meshes, has good convergence properties, and is computationally efficient. The new scheme, called DD-L2-ENO, uses a data-dependent weighted least-squares reconstruction with a fixed stencil. The weights are chosen to strongly emphasize smooth data in the reconstruction, satisfying the weighted ENO conditions of Liu, Osher, and Chan. Because DD-L2-ENO is designed in the framework ofk-exact reconstruction, existing techniques for implementing such reconstructions on arbitrary meshes can be used. The scheme allows graceful degradation of accuracy in regions where insufficient smooth data exists for reconstruction of the requested high order. Similarities with and differences from WENO schemes are highlighted. The asymptotic behavior of the scheme in reconstructing smooth and piecewise smooth functions is demonstrated. DD-L2-ENO produces uniformly high-order accurate reconstructions, even in the presence of discontinuities. Results are shown for one-dimensional scalar propagation and shock tube problems. Encouraging preliminary two-dimensional flow solutions obtained using DD-L2-ENO reconstruction are also shown and compared with solutions using limited least-squares reconstruction.

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