Time splitting in DNS
 

Orszag + Patterson ["Num. Sim. of 3D Homogeneous Isotropic Turbulence", Phys. Rev. Lett. (1972)] solve homogeneous isotropic turbulence via a spectral method.

They combine the solution of the Poisson equation with the Navier-Stokes equation so that the solution automatically satisfies continuity. Thus, they don't use a time-splitting algorithm.

Yet, spectral methods for inhomogeneous flows have to use a time splitting (or coupled) method.

I'm assuming that they are able to enforce continuity analytically thru their equation (2) because the flow is homogenous-isotropic. However, I don't completely understand why this can't be done for inhomogeneous flows. Is it because of the "non-periodic" boundary conditions that a time splitting method has to be used for non-homogeneous flows.

Thanks.