  In this paper, we develop new techniques for solving the large, coupled
linear systems that arise from fully implicit Runge-Kutta 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 Runge-Kutta
(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 Runge-Kutta stages simultaneously. Numerical experiments are
performed on the Navier-Stokes equations using Euler vortex and 2D and 3D NACA
airfoil test cases in serial and in parallel settings. The fully implicit Radau
IIA Runge-Kutta methods compare favorably with equal-order DIRK methods in
terms of accuracy, number of GMRES iterations, number of matrix-vector
multiplications, and wall-clock time, for a wide range of time steps.

Numerical Analysis