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Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy andstability properties as time integration schemes, but high-order IRK methodsare not commonly used in practice with large-scale numerical PDEs because ofthe difficulty of solving the stage equations. This paper introduces atheoretical and algorithmic framework for solving the nonlinear equations thatarise from IRK methods (and discontinuous Galerkin discretizations in time)applied to nonlinear numerical PDEs, including PDEs with algebraic constraints.Several new linearizations of the nonlinear IRK equations are developed,offering faster and more robust convergence than the often-consideredsimplified Newton, as well as an effective preconditioner for the true Jacobianif exact Newton iterations are desired. Inverting these linearizations requiressolving a set of block 2x2 systems. Under quite general assumptions, it isproven that the preconditioned 2x2 operator's condition number is bounded by asmall constant close to one, independent of the spatial discretization, spatialmesh, and time step, and with only weak dependence on the number of stages orintegration accuracy. Moreover, the new method is built using the samepreconditioners needed for backward Euler-type time stepping schemes, so can bereadily added to existing codes. The new methods are applied to severalchallenging fluid flow problems, including the compressible Euler and NavierStokes equations, and the vorticity-streamfunction formulation of theincompressible Euler and Navier Stokes equations. Up to 10th-order accuracy isdemonstrated using Gauss IRK, while in all cases 4th-order Gauss IRK requiresroughly half the number of preconditioner applications as required by standardSDIRK methods.

Fast solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part II: nonlinearities and DAEs