A Fast High-Order Solver For Stratified Flows on Massively Multi-cored Architectures

The simulation of high Reynolds number stratified flows has important oceanographic and atmospheric applications. In many flows the presence of strongly stratified turbulence leads to the formation of thin horizontal regions of high shear, necessitating the use of high resolution non-uniform grids in the vertical. Additionally, long-time integrations are required as fluid motions persist for much greater durations than their unstratified counterparts. We will present a fast, high-order solver for stratified flows utilizing a Fourier pseudo-spectral method in the horizontal and a spectral element discretization in the vertical. An IMEX time-splitting scheme is used, requiring the solution of several 1D Helmholtz equations during each time-step. Ultraspherical polynomial basis functions in combination with static condensation subsequently result in a large number of small tridiagonal systems, and hence an algorithm that is as inexpensive as second-order finite difference schemes. Many levels of coarse and fine grain parallelism are exploited to achieve optimal performance on massively multi-cored processors, which are prevalent in today’s high-performance computing environment. Simulations of a stratified turbulent wake are used as a numerical example.