To fully understand the bifurcation-theoretic structure of solutions to a set of nonlinear evolution equations, three types of information are desirable. The first, and most often used in fluid dynamics, is the time evolution of the system from various initial conditions. These constitute the physically realizable phenomena to be explained. The second type of information is the set of steady-state solutions. For equations describing medium and high Reynolds number hydrodynamic systems, the number of steady states can be vast; however it is useful to obtain as much of the picture as possible, especially concerning unstable steady states. The third type of information is the eigenspectrum of the steady states. Leading eigenvalues are associated with transitions and loss of stability.
These three types of information are usually obtained by separate analyses, involving separate codes and researchers. Here we propose a unified computational approach to the three types of calculation, all based on a single time-dependent code and using the same set of low-level routines. We can readily transfer flow fields between each of the three computational tools: Time-dependent simulations can be used to generate initial states for branch continuation. Unstable steady states, possibly perturbed by the addition of eigenvectors, can serve as initial conditions for time evolution. Suspected bifurcations can be confirmed by linear stability analysis.
A key step is to use iterative Krylov techniques to invert and diagonalize the Jacobian matrix, which is too large to store or to transform directly. At the low and moderate Reynolds numbers for which bifurcation-theoretic analysis is most meaningful, convergence of Krylov techniques is greatly accelerated by preconditioning the Jacobian matrix with the Stokes operator.