Implicit high order discontinuous Galerkin method for compressible laminar and turbulent flow simulation

In order to improve the stability of the implicit higher order discontinuous Galerkin (DG)method, an implicit GMRES method based on analytic exact Jacobian matrix is developed for simulating compressible laminar and turbulent flow. In the process of the GMRES, Jacobians of inviscid and viscous flux, which are solved by the chain rule, are used for the LU-SGS preconditioning and GMRES matrix-vector product. The source of the Spalart-Allmaras turbulence model is modified to avoid unphysical solution that occurs at implicit derivative. The developed method is validated by benchmark laminar and turbulent flow cases. The results show that the implicit GMRES method based on exact Jacobian possesses significant advantages in reducing computational cost and improving the stability of the implicit high order discontinuous Galerkin method.