A third order gas-kinetic scheme for unstructured grid

In order to study the high order gas-kinetic scheme on unstructured grid, we combine the third order gas-kinetic flux solver with the compact least-square reconstruction (CLS) method together. The CLS method can achieve arbitrary high order compact reconstruction using the stencil from the whole computational domain implicitly. A large sparse linear system resulted from the CLS reconstruction method is solved by applying the generalized minimal residual algorithm (GMRES). To accelerate the convergence, the Reverse-Cuthill–McKee (RCM) algorithm and the incomplete lower–upper (ILU) factorization method are implemented. Different from the traditional flux solver, the gas-kinetic scheme developed by Xu is of high spatial and temporal accuracy. Applying the second order Taylor expansion of the initial gas distribution function and equilibrium state at cell interface, the flux solver can be extended to third order accuracy directly. The accuracy of present method is validated by several numerical cases such as the advection of density perturbation problem, isotropic vortex propagation problem, Sod shock–wave problem, Lax shock tube test case, Shu–Osher problem, shock–vortex interaction, lid-driven cavity flow, and flat plate boundary layer. The advantages of this high order gas-kinetic scheme are exhibited in some benchmarks including incompressible flow and supersonic compressible flow, inviscid flow and viscous flow.