Rates of Convergence for Viscous Splitting of the Navier-Stokes Equations

Viscous splitting algorithms are the underlying design principle for many numerical algorithms which solve the Navier-Stokes equations at high Reynolds number. In this work, error estimates for splitting algorithms are developed which are uniform in the viscosity $\nu$ as it becomes small for either two- or three-dimensional fluid flow in all of space. In particular, it is proved that standard viscous splitting converges uniformly at the rate $C\nu \Delta t$, Strang-type splitting converges at the rate $C\nu {(\Delta t)^2}$, and also that solutions of the Navier-Stokes and Euler equations differ by $C\nu$ in this case. Here C depends only on the time interval and the smoothness of the initial data. The subtlety in the analysis occurs in proving these estimates for fixed large time intervals for solutions of the Navier-Stokes equations in two space dimensions. The authors derive a new long-time estimate for the two-dimensional Navier-Stokes equations to achieve this. The results in three space dimensions are valid for appropriate short time intervals; this is consistent with the existing mathematical theory.