A fourth-order-accurate and compact finite volume method (FVM) on curvilinear grids

To our knowledge, it is difficult to apply the accurate and compact FVM to curvilinear grids because of the difficulty in calculating integral approximation accurately on curvilinear grids. With the coordinate transform, we derive the equations for calculating fourth-order-accurate cell-averaged variables and interface-averaged variables so as to solve the integral approximation problem in the FVM and the curvilinear grid application problems. We use the fourth-order Padé compact scheme to carry out the spatial discretization of the Euler equations. We derive an integral-type high-order compact filtering equation to replace the artificial dissipation in order to converge the calculation in the time marching process. Finally, we give two numerical simulation examples to verify the correctness and effectiveness of our method. The simulation results, given in Figs. 2 through 6 and Table 1, show preliminarily that: (1) the calculation of the flow over a cylinder and the Ringleb flow with our method can reach the fourth-order accuracy; (2) our method can accomplish high-order integral approximation and solve the curvilinear grid application problems.