The application of upwind meshless approximation finite volume method to the convection dominated problems on the unstructured and deformed meshes

Abstract

The primary objective of this research is to develop a finite volume method that can incorporate the most widely recognized upwind schemes using meshless techniques on an unstructured mesh. The recently developed Meshless Approximation Finite Volume Method (MAFVM) is expected to provide a solution to the convection–diffusion equation with an extremely high Pélect number, regardless of the type or quality of mesh, even when dealing with highly distorted meshes. By incorporating the local compactly supported radial basis functions differential quadrature method (LCRBFDQ), a meshless approximation technique, the field variables, flux, and gradient are directly approximated using information from neighboring scattered nodes. Building upon this concept, we propose two meshless approximation approaches for computing the flux and gradient within the framework of the finite volume method. Elliptic type problem on regular mesh is solved using the proposed MAFVM by different flux computation methods. Parameters for MAFVM that provide accurate and robust results are suggested. Next standard linear and second order upwind schemes as well as the well-known total variation diminishing (TVD) methods are integrated into the proposed MAFVM, using the developed flux calculation method to solve convection–diffusion problem with very high Pélect number on standard or highly deformed mesh. Accurate and robust results can be observed which show that the current MAFVM can provide an alternative and better way to deal with strong convection problem on meshes of arbitrary types by the proposed meshless flux computation methods.