Stage-parallel fully implicit Runge-Kutta solvers for discontinuous Galerkin fluid simulations

In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit RungeKutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit RungeKutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the RungeKutta stages simultaneously. Numerical experiments are performed on the NavierStokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA RungeKutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrixvector multiplications, and wall-clock time, for a wide range of time steps.