  A high order finite difference method is proposed for unstructured meshes to
simulate compressible inviscid/viscous flows with/without discontinuities. In
this method, based on the strong form equation, the divergence of the flux on
each vertex is computed directly from fluxes nearby by means of high order
least-square. In order to capture discontinuities, numerical flux of high order
accuracy is calculated on each edge and serves as supporting data of the
least-square computation of the divergence. The high accuracy of the numerical
flux depends on the high order WENO interpolation on each edge. To reduce the
computing cost and complexity, a curvlinear stencil is assembled for each edge
so that the economical one-dimensional WENO interpolation can be applied. With
the derivatives introduced, two-dimensional Hermite interpolation on a
curvilinear stencil is applied to keep the stencil compact and avoids using
many supporting points. In smooth region, the Hermite least-square 2D
interpolation of 5 nodes is adopted directly to achieve the fifth order
accuracy. Near a discontinuity, three values obtained by means of least-square
2D interpolation of 3 nodes, are weighted to obtain one value of the second
order accuracy. After obtaining the flow states on both sides of the middle
point of an edge, numerical flux of high order accuracy along the edge can be
calculated. For inviscid flux, analytical flux on vertices and numerical flux
along edges are used to compute the divergence. While for viscous flux, only
analytical viscous flux on vertices are used. The divergence of the fluxes and
their derivatives on each vertex are used to update the conservative variables
and their derivatives with an explicit Runger-Kutta time scheme. Several
canonical numerical cases were solved to test the accuracy and the capability
of shock capturing of this method.
