Recent development of high order DG/FV hybrid methods

A concept of ‘static reconstruction’ and ‘dynamic reconstruction’ had been introduced for higher-order (third-order and higher) numerical methods in our previous work. Based on this concept, a class of DG/FV hybrid methods had been developed for the scalar equations and Euler/NS equations on triangular and Cartesian/triangular hybrid grids. In this paper, the recent progress of the DG/FV hybrid methods was presented. The basic idea of ‘hybrid reconstruction’, the procedure of solving NS equations with BR2 approach, and the implicit algorithm were reviewed briefly. And then the dissipative and dispersive property, as well as the stability, of the DG/FV hybrid schemes were analyzed. In order to show the high efficiency in the term of CPU time of the present DG/FV hybrid schemes, the computational costs were discussed and compared with the corresponding DG methods. The numerical accuracy was validated by some typical test cases of viscous flow, including the Couette flow, laminar flow in a square, compressible mixing layer problem, turbulent flows by RANS equations with S-A turbulent model over a flat plate and over NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired order of accuracy, and they can capture the flow structure accurately. Qualitative analysis and numerical applications demonstrate that they can reduce the CPU time greatly (up to 40%) comparing with the traditional DG method with the same order of accuracy. Meanwhile, the implicit algorithm can accelerate the convergence history obviously, one to two orders faster than the explicit RungeKutta method.