# Block preconditioned implicit Runge-Kutta methods for the incompressible Navier-Stokes equations

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## Abstract

We examine block preconditioners for time-dependent incompressible Navier-Stokes problems. In some time-dependent problems, explicit time-stepping methods can require much smaller time steps for stability than are needed for reasonable accuracy. This leads to taking many more time steps than would otherwise be needed. With implicit time-stepping methods, we can take larger steps, but at the price of needing to solve large linear systems at each time step. We consider implicit Runge-Kutta (IRK) methods. Suppose our PDE has been linearized and discretized with N degrees of freedom. Using an s-stage IRK method leads to an sN by sN linear system that must be solved at each time step. These linear systems are block s by s systems, where each block is N by N. We investigate preconditioners for such systems, where we take advantage of the fact that each subblock is related to a linear system from the steady-state fluid flow equations, for which there are several effective preconditioners.

One such preconditioner is the least squares commutator (LSC) preconditioner, which has a block triangular form that is based on an LU factorization of the steady-state linear system and uses an approximation to the Schur complement in the (2,2) block. We investigate the performance of the LSC preconditioner applied to the blocks of the IRK system which must be solved at each time step of the time-dependent problem.

We explore a variety of IRK methods of different stages for solving the time-dependent incompressible Navier-Stokes equations. We find that Radau methods are particularly effective because of their stability properties and stiff decay. Because the Navier-Stokes equations are differential algebraic equations (DAE), which are infinitely stiff, these stability properties are highly desirable.