An implicit discontinuous Galerkin method for the unsteady compressible Navier–Stokes equations

A high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier–Stokes equations. The spatial discretization is carried out using a high order discontinuous Galerkin method, where polynomial solutions are represented using a Taylor basis. A second order implicit method is applied for temporal discretization to the resulting ordinary differential equations. The resulting non-linear system of equations is solved at each time step using a pseudo-time marching approach. A newly developed fast, p-multigrid is then used to obtain the steady state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady subsonic viscous flow problems. The numerical results obtained indicate that the use of this implicit method leads to significant improvements in performance over its explicit counterpart, while without significant increase in memory requirements.

► An implicit discontinuous Galerkin method is developed for the compressible Navier–Stokes equations.
► The resulting non-linear equations are solved using a pseudo-time marching approach.
► A p-multigrid method is used to obtain the steady state solution to the pseudo-time system.
► This implicit method provides significant improvement in performance over its explicit counterpart.