1 research outputs found
Fast solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part II: nonlinearities and DAEs
Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and
stability properties as time integration schemes, but are rarely used in
practice with large-scale numerical PDEs because of the difficulty of solving
the stage equations. This paper introduces a theoretical and algorithmic
framework for solving the nonlinear equations that arise 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-considered simplified Newton, as well as an
effective preconditioner for the true Jacobian if exact Newton iterations are
desired. Inverting these linearizations requires solving a set of block 2x2
systems. Under quite general assumptions, it is proven that the preconditioned
2x2 operator has a condition number of ~O(1), independent of the spatial
discretization, and with only weak dependence on the number of stages or
integration accuracy. Moreover, the new method is built using the same
preconditioners needed for backward Euler-type time stepping schemes, so can be
readily added to existing codes. The new methods are applied to several
challenging fluid flow problems, including the compressible Euler and Navier
Stokes equations, and the vorticity-streamfunction formulation of the
incompressible Euler and Navier Stokes equations. Up to 10th-order accuracy is
demonstrated using Gauss IRK, while in all cases 4th-order Gauss IRK requires
roughly half the number of preconditioner applications as required by standard
SDIRK methods.Comment: 30 page