An implicit, reconstructed discontinuous galerkin method for the unsteady compressible navier-stokes equations on 3D hybrid grids

An implicit, third-order, reconstructed discontinuous Galerkin method, namely RDG (P1P2), is presented for time-accurate solutions to the compressible Navier-Stokes equations on 3D hybrid grids. The spatial discretization is carried out using a Taylor-basis discontinuous Galerkin method, in which a quadratic polynomial (P2) solution is reconstructed via a WENO (P1P2) reconstruction scheme from the underlying linear polynomial discontinuous Gakerkin (P1) solution in each cell for evaluating the fluxes. A series of Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta schemes (termed ESDIRK) are applied for temporal discretization to the resulting ordinary differential equations. The resulting non-linear system of equations at each stage is solved using an approximate Newton's method, in which an LU-SGS preconditioned GMRES solver is applied for the solution to the linear system of equations. The developed code is applied to compute a series of benchmark test cases, including the implicit large eddy simulation of a turbulent lid driven cavity. The numerical results indicate that the use of ESDIRK schemes leads to remarkable improvement in solution efficiency and temporal accuracy over its explicit counterpart. In addition, this implicit RDG (P1P2) method requires much less storage and computing time than the implicit DG (P2) method, resulting in a fast, third-order implicit discontinuous Galerkin method for computing unsteady flow problems.