Time Marching Kernel Approximated PDE Solutions for Meshfree Computational Fluid Dynamics

This paper will address the problem of time marching function approximated solutions inherent in emerging meshfree Computational Fluid Dynamics (CFD) solution techniques. The numerical solutions of partial differential equations (PDEs) of CFD has been dominated by either finite difference methods (FDM), finite element methods (FEM), and finite volume methods (FVM). These methods can be derived from the assumptions of the Taylor expansion based local interpolation schemes and they require a mesh to support the local approximation. The problem is that in complex shaped domains, the construction of the mesh is a non-trivial problem. Typically with these methods, only the function is continuous across meshes, but not its partial derivatives. The difficulties of mesh construction and discontinuous derivatives has led to the development of mesh independent methods or meshfree (MF). These new meshfree methods represent the next generation of CFD solvers as they mature. In these methods the local function approximation method is independent of the mesh (or design points) of the geometric domain in which a solution is sought. In this paper we investigate the approximation of the local function by the kernel based statistical method of Nadaraya and Watson (NW). We show how the approximated solution in an arbitrary mesh can be matched in time to obtain the steady state and/or time dependent solution of the PDE.