PSH3D fast Poisson solver for petascale DNS

Direct numerical simulation (DNS) of high Reynolds number, ${Re} \ge O(10^5)$, turbulent flows requires computational meshes $\ge O(10^{12})$ grid points, and, thus, the use of petascale supercomputers. DNS often requires the solution of a Helmholtz (or Poisson) equation for pressure, which constitutes the bottleneck of the solver. We have developed a parallel solver of the Helmholtz equation in 3D, PSH3D. The numerical method underlying PSH3D combines a parallel 2D Fast Fourier transform in two spatial directions, and a parallel linear solver in the third direction. For computational meshes up to $8192^{3}$ grid points, our numerical results show that PSH3D scales up to at least 262k cores of Cray {XT5} (Blue Waters). PSH3D has a peak performance 6$\times$ faster than 3D FFT-based methods when used with the `partial-global' optimization, and for a $8192^3$ mesh solves the Poisson equation in 1~sec using 128k cores. Also, we have verified that the use of PSH3D with the `partial-global' optimization in our DNS solver does not reduce the accuracy of the numerical solution of the incompressible Navier-Stokes equations.