### A FFT Preconditioning Technique for the Solution of Incompressible Flow with Fractional Step Methods on Graphic Processing Units

*Mario A. Storti, Rodrigo R. Paz, Lisandro D. Dalcín, Santiago Costarelli*

#### Abstract

The resolution of Computational Fluid Dynamics (CFD) problems on Graphic Processing Units (GPU’s) requires of specialized algorithms due to the particular hardware architecture of these devices. Algorithms that fall in the category of cellular automata (CA) are the best ﬁtted, for instance explicit Finite Volume or Finite Element methods. But in the case of incompressible ﬂow it is not possible to develop a pure explicit algorithm, due to the essentially non-local character of the incompressibility condition. In this case the algorithms that are closer to an explicit approach, are segregated algorithms, like the Fractional Step Method. In this algorithms the more time consuming stage is (asymptotically for large problems) the solution of the Poisson’s equation for pressure. A common choice for it’s solution is the IOP (Iterated Orthogonal Projection) method, which requires a series of solutions on the complete mesh. In this work a variant of the IOP, called Accelerated Global Preconditioning (AGP), is proposed. It is based on using a Preconditioned Conjugate Gradient (which is an accelerated iterative method, in contrast with the stationary scheme used in IOP) for the pressure on the ﬂuid, and preconditioning with the solution on the global domain (ﬂuid and solid). Of course, solving the problem on the global domain represents more computational work than solving the problem only in the ﬂuid, but this can be faster in a structured mesh if a fast solvers as Multigrid or Fast Fourier Transform (FFT) is used. The main advantage of AGP over IOP is that it is an accelerated solver, whereas the IOP is stationary. In addition AGP iterates only on pressure, whereas IOP iterates on both pressure and velocity.

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