A third order accurate cellwise relaxation implicit Discontinuous Galerkin (DG) scheme for RANS simulations using unstructured hybrid meshes is presented. A scalar parabolic equation is first examined to clarify what is really important in construction of implicit matrix to keep its diagonal dominance for the third and fourth order cellwise relaxation implicit DG schemes. In addition, discussions are given to approximated construction of implicit matrix for reducing computational cost. Then, the third order accurate cellwise relaxation implicit DG scheme for RANS simulations is successfully developed utilizing the expertise learned in the study of solving the parabolic equation. Superior spatial accuracy of the third order accurate cellwise relaxation implicit DG scheme for RANS simulations, while retaining reasonable convergence properties, is demonstrated for typical aerospace applications.
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