Least squares Kinetic Upwind Mesh-free Method

Least squares kinetic upwind mesh-free (LSKUM) method has been the subject of research over twenty years in our research group. LSKUM method requires a cloud (W) of points or nodes and connectivity N(P 0 ) for every 0 P ˛W. The connectivity of P 0 is a set of neighbours 0 () i PN P ˛ of P 0 . The cloud can be a simple cloud, Cartesian cloud or chimera cloud or can be obtained rapidly using advancing front method. The discrete approximation to spatial derivatives was obtained using of least squares and it can be made accurate using defect correction method. The LSKUM first operates on the Boltzmann level and then passes on to Euler or Navier-Stokes level by taking suitable moments (so called y moments) of the Boltzmann equation of kinetic theory of gases. The upwinding in LSKUM method is enforced by stencil or connectivity splitting based on the signs of v 1 , v 2 in 2-D and v 1 , v 2 , v 3 in 3-D. This leads to split fluxes encountered in Kinetic Flux Vector Splitting (KFVS) method. The higher-order accurate LSKUM method can be made more efficient using entropy variables, thus leading to q-LSKUM method. Lastly, boundary conditions are implemented using specular reflection model on the wall (KCBC method) and by using kinetic outer boundary condition (KOBC) method for a point on the outer boundary.