The lower-upper symmetric Gauss-Seidel (LU-SGS) implicit relaxation has been widely used because it has the merits of less dependency on grid topology, low numerical complexity and modest memory requirements. In original LU-SGS scheme, the implicit system matrix is constructed based on the splitting of convective flux Jacobian according to its spectral radius. Although this treatment has the merit of reducing computational complexity and helps to ensure the diagonally dominant property of the implicit system matrix, it can also cause serious distortions on the implicit system matrix because too many approximations are introduced by this splitting method if the contravariant velocity is small or close to sonic speed. To overcome this shortcoming, an improved LU-SGS scheme with a hybrid construction method for the implicit system matrix is developed in this paper. The hybrid way is that: on the cell faces having small contravariant velocity or transonic contravariant velocity, the accurate derivative of the convective flux term is used to construct more accurate implicit system matrix, while the original Jacobian splitting method is adopted on the other cell faces to reduce computational complexity and ensure the diagonally dominant property of the implicit system matrix. To investigate the convergence performance of the improved LU-SGS scheme, 2D and 3D turbulent flows around the NACA0012 airfoil, RAE2822 airfoil and LANN wing are simulated on hybrid unstructured meshes. The numerical results show that the improved LU-SGS scheme is significantly more efficient than the original LU-SGS scheme. © 2012 Chinese Journal of Aeronautics.
Wang, G., Jiang, Y., & Ye, Z. (2012). An improved LU-SGS implicit scheme for high Reynolds number flow computations on hybrid unstructured mesh. Chinese Journal of Aeronautics, 25(1), 33–41. https://doi.org/10.1016/S1000-9361(11)60359-2