A Diagonally Implicit Runge-Kutta Method for the Discontinuous Galerkin solutions of the Navier-Stokes Equations

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A high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier-Stokes equations on arbitrary grids. The spatial discretization is carried out using a high order discontinuous Galerkin method, where polynomial solutions are represented using a Taylor basis. A diagonally implicit RungeKutta method is applied for temporal discretization to the resulting ordinary differential equations. The resulting nonlinear system of equations is solved at each stage 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 viscous flow problems. The numerical results obtained indicate that the use of this implicit method leads to orders of improvements in performance over its explicit counterpart, while without significant increase in memory requirements.