DEVELOPMENT OF HIGH-ORDER FINITE VOLUME METHODS ON UNSTRUCTURED GRIDS AND THEIR APPLICATIONS

Abstract

To address some of the bottleneck problems of the existing high-order methods, two straightforward mesh-free scheme-based high-order finite volume (FV) methods on unstructured grids are developed in this thesis for simulation of various flow problems. Under the basic framework of the unstructured finite volume method, a high-order polynomial based on Taylor series expansion is constructed within every control cell to approximate the solution function. The unknown coefficients in the Taylor series expansion, i.e., the various derivatives, are approximated by the functional values at the cell centers of the considered cell and its neighboring cells using the mesh-free least square-based finite difference (LSFD) and local radial basis function-based differential quadrature (LRBFDQ) schemes. The developed high-order LSFD-FV and LMQDQ-FV methods are straightforward, user-friendly, easy to implement and flexible in modelling complex flow problems on arbitrary grids.