Ramifications of Implicit Runge-Kutta Time Integration Scheme

Implicit Runge-Kutta methods are more and more coming into the focus of algorithmic research to increase efficiency when numerically solving the Navier-Stokes equations. In this contribution, three variations of Implicit Runge-Kutta methods proposed so far are investigated: first, for future parallelization efforts a red/black Gauss-Seidel iteration procedure is introduced, and compared to the standard lexicographic Gauss-Seidel iteration similar performance is shown. Second, the implicit Left-Hand-Side operator is simplified such that the method becomes formally independent of the Right-Hand-Side spatial discretization, and by careful formulation similar computation times as with the full first order Jacobian are established. Third, the sensitivity of computation time with respect to the number of global sub-iterations is reduced by introducing local sub-iterations based on the size of the actual residual. Computation of two-dimensional compressible and incompressible, laminar and turbulent airfoil flows using the resulting methods confirms the applicability to different flow cases.