High-order implicit discontinuous Galerkin schemes for unsteady compressible Navier–Stokes equations

Efficient solution techniques for high-order temporal and spatial discontinuous Galerkin (DG) discretizations of the unsteady Navier–Stokes equations are developed. A fourth-order implicit Runge–Kutta (IRK) scheme is applied for the time integration and a multigrid preconditioned GMRES solver is extended to solve the nonlinear system arising from each IRK stage. Several modifications to the implicit solver have been considered to achieve the efficiency enhancement and meantime to reduce the memory requirement. A variety of time-accurate viscous flow simulations are performed to assess the resulting high-order implicit DG methods. The designed order of accuracy for temporal discretization scheme is validate and the present implicit solver shows the superior performance by allowing quite large time step to be used in solving time-implicit systems. Numerical results are in good agreement with the published data and demonstrate the potential advantages of the high-order scheme in gaining both the high accuracy and the high efficiency.