Operator splitting methods for the Navier-Stokes equations

Abstract

The properties of two algorithms for the solution of the incompressible Navier-Stokes equations are examined. These algorithms are representative of the two prevalent splitting philosophies used to split the Navier-Stokes equations. The properties of the algorithms are determined by Fourier analysis of a model problem. It is shown that splittings resulting in Stokes subproblems have desirable dissipative properties which are lacking in the velocity-pressure splittings.

Algorithms for solving subproblems resulting from the splitting of the Navier-Stokes equations are introduced and compared to existing methods. The new methods are based on a splitting of the Stokes matrix and subsequent orthogonal projection of the intermediate velocity onto a divergence free subspace. Algorithms based on several splittings are examined and compared to the conjugate gradient Uzawa algorithm. The split matrix scheme is accelerated by application of a Krylov subspace outer iteration. The accelerated scheme uses the conjugate gradient method if the iteration matrix is symmetric and positive definite and the Gmres algorithm if not. The accelerated scheme performed as well as or better than the Uzawa scheme.

The nonlinear subproblem is preconditioned via an approximate Jacobian secant method. In this technique the secant method is accelerated by the application of a nonlinear Krylov method to the nonlinear system of equations which result in the application of Newtons method. The number of function evaluations required to solve the nonlinear system was greatly reduced by the application of the preconditioner.