Compact upwind schemes on adaptive octrees

Compact high-order upwind schemes using reconstruction from cell-averages are derived for application with the compressible three-dimensional Navier–Stokes equations. An adaptive-octree mesh, combined with the Adams–Bashforth–Moulton family of predictor–corrector schemes, provides a conservative high-order time-integration platform supporting localized h-refinement and timestep sub-cycling. Numerical examples for smooth flows demonstrate the improvement over explicit upwind schemes and formal accuracy of the schemes, as well as the behavior in wall-bounded regions, and the resolution of a broad wavenumber spectrum.