LONG-TERM RIGOROUS NUMERICAL INTEGRATION OF NAVIER-STOKES EQUATION BY NEWTON-GMRES ITERATION

The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numer- ical integration of an orbit on the unstable manifold of a periodic solution. This algorithm is matrix-free and em- ploys a combination of the Newton-Raphson method and the Krylov subspace method. Moreover, the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integra- tion. The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata~s lower-branch steady solution of plane Couette flow, which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.