A high-order hybrid finite difference–finite volume approach with application to inviscid compressible flow problems: A preliminary study

A class of hybrid finite difference–finite volume (FD–FV) operators is recently developed as building blocks to solve one dimensional hyperbolic conservation laws when the solutions are smooth. This method differs from conventional finite difference (FD) or finite volume (FV) schemes in that both nodal values and cell-averaged values are considered as dependent variables and they are evolved in time. Under this framework, the 1D FD–FV methods: (1) are numerically conservative for cell averages; (2) have straightforward extension to high-order accuracy; and (3) have superior spatial accuracy property compared to most conventional FD or FV methods.This work extends the FD–FV approach in two aspects. The first extension is a WENO-type stabilization to enhance the nonlinear stability of sample high-order 1D FD–FV operators. In particular, numerical results show that when the solutions are smooth, the optimal order of accuracy (fifth-order) is achieved by the stabilized fourth-order FD–FV method; and it is also capable to handle problems with strongly discontinuous solutions.The second part of the paper extends a second-order FD–FV method to two-dimensional smooth problems. Both Cartesian grids and unstructured (triangular) grids are considered. In multiple dimensions, there are different choices of the collocation points of the nodal values, and they lead to different FD–FV schemes. This work develops a node-centered FD–FV scheme and an edge-centered FD–FV scheme on each type of grids, and their numerical performance are assessed and compared by solving benchmark flow problems with smooth solutions. In particular, the numerical examples confirm that the superior spatial accuracy property of the 1D FD–FV operators carries to two space dimensions on Cartesian grids. The present work focuses on two space dimensions, but the methodology extends naturally to three-dimensional Cartesian grids and tetrahedral grids.