基于非结构/混合网格的高阶精度DG/FV混合方法研究进展

Recent development of high order DG/FV hybrid methods

DG／FV 混合方法因其具有紧致、易于推广获得高阶格式及相比同阶精度 DG 方法计算量、存储量小等优点，自提出以来已成功应用于一维、二维标量方程和 Euler／N-S 方程的求解。综述了 DG／FV 混合方法的研究进展，重点介绍了 DG／FV 混合方法的空间重构算法、针对 RANS 方程的求解方法、隐式时间离散格式、数值色散耗散及稳定性分析、计算量理论分析，并给出了系列粘性流算例的计算结果，包括用于验证混合方法数值精度的库埃特流，以及方腔流、亚声速剪切层、低速平板湍流、NACA0012翼型湍流绕流等。数值计算结果表明 DG／FV 混合方法达到了设计的精度阶，且相比同阶 DG 方法计算量减少约40％，而隐式方法能大幅提高定常流的收敛历程，较显式 Runge-Kutta 的收敛速度提高1～2个量级。

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 trian-gular 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 proper-ty,as well as the stability,of the DG/FV hybrid schemes were analyzed.In order to show the high efficien-cy 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.