  In this paper, a third-order compact gas-kinetic scheme (GKS) on unstructured
tetrahedral mesh is constructed for the compressible Euler and Navier-Stokes
solutions. The time-dependent gas distribution function at a cell interface is
used to calculate the fluxes for the updating the cell-averaged flow variables
and to evaluate the time accurate cell-averaged flow variables as well for
evolving the cell-averaged gradients of flow variables. With the accurate
evolution model for both flow variables and their slopes, the quality of the
scheme depends closely on the accuracy and reliability of the initial
reconstruction of flow variables. The reconstruction scheme becomes more
challenge on tetrahedral mesh, where the conventional second-order unlimited
least-square reconstruction can make the scheme be linearly unstable when using
cell-averaged conservative variables alone with von Neumann neighbors.
Benefiting from the evolved cell-averaged slopes, on tetrahedral mesh the GKS
is linearly stable from a compact third-order smooth reconstruction with a
large CFL number. In order to further increase the robustness of the high-order
compact GKS for capturing discontinuous solution, a new two-step
multi-resolution weighted essentially non-oscillatory (WENO) reconstruction
will be proposed. The novelty of the reconstruction includes the following.
Firstly, it releases the stability issue from a second-order compact
reconstruction through the introduction of a pre-reconstruction step. Secondly,
in the third-order non-linear reconstruction, only one more large stencil is
added beside those in the second-order one, which significantly simplifies the
high-order reconstruction. The proposed third-order scheme shows good
robustness in high speed flow computation and favorable mesh adaptability in
cases with complex geometry.
