Application of Krylov subspace methods in fluid dynamics

A robust, fast, and powerful technique, based on Krylov subspace methods, is presented for solving large nonlinear equations of the form F(u) = O. The main methods investigated are (a) a standard Newton approach coupled with a direct or iterative sparse solver and (b) a Jacobian-free Krylov subspace Newton method. The methods are applied to fluid dynamics problems. In all tested cases, the Jacobian-free Krylov subspace methods based on a nonlinear Generalized Minimum Residual (GMRES) technique show better performance when compared with the standard Newton technique. The importance of selective preconditioners for improving the convergence is demonstrated. The two-dimensional driven cavity problem is solved for Reynolds number 3000, starting from the zero initial guess, using the nonlinear GMRES technique with the line search backtracking.