A third-order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time-accurate solution of the compressible Navier-Stokes equations

A space and time third-order discontinuous Galerkin method based on a Hermite weighted essentially non-oscillatory reconstruction is presented for the unsteady compressible Euler and Navier-Stokes equations. At each time step, a lower-upper symmetric Gauss-Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge-Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third-order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third-order discontinous Galerkin methods, and less computing time than the lower-order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems.