A Continuation and Bifurcation Technique for Navier–Stokes Flows

An efficient numerical bifurcation and continuation method for the Navier–Stokes equations in cylindrical geometries is presented and applied to a nontrivial fluid dynamics problem, the flow in a cylindrical container driven by differential rotation. The large systems that result from discretizing the Navier–Stokes equations, especially in regimes where inertia is important, necessitate the use of iterative solvers which in turn need preconditioners. We use incomplete lower–upper decomposition (ILU) as an effective preconditioner for such systems and show the significant gain in efficiency when an incomplete LU of the full Jacobian is used instead of using only the Stokes operator. The computational cost, in terms of CPU time, grows with the size of the system (i.e., spatial resolution) according to a power law with exponent around 1.7, which is very modest compared to direct methods, indicating the appropriateness of the schemes for large nonlinear partial differential equation problems.