Improving Godunov-type reconstructions for simulation of vortex-dominated flows

A systematic Fourier accuracy analysis is performed to examine the numerical diffusion inherent in a Godunov-type reconstruction, including both the reconstruction of the solution within each cell and the computation of the derivative terms of the reconstruction. It is found that compared with the more popular fifth-order polynomial fit of the interface values, a piecewise quadratic reconstruction of the solution with more accurate slope and curvature, especially those computed by compact difference schemes, is much less dissipative. Therefore, further given in the paper is a general framework to make a piecewise quadratic reconstruction free of numerical oscillations around the shocks. The improved accuracy and robustness of the resulting Godunov-type schemes for simulation of vortex-dominated flows are demonstrated with the numerical results of several carefully selected cases, including vortex convection and shock–vortex interaction.