A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet

Abstract

a b s t r a c tModern hardware architectures such as GPUs and manycore processors are characterised by an abundance of compute capability relative to memory bandwidth. This makes them well-suited to solving temporally explicit and spatially compact discretisations of hyperbolic conservation laws. However, classical pressure-projection-based incompressible Navier-Stokes formulations do not fall into this category. One attractive formulation for solving incompressible problems on modern hardware is the method of artificial compressibility. When combined with explicit dual time stepping and a high-order Flux Reconstruction discretisation, the majority of operations can be cast as compute bound matrix-matrix multiplications that are well-suited for GPU acceleration and manycore processing. In this work, we develop a high-order cross-platform incompressible Navier-Stokes solver, via artificial compressibility and dual time stepping, in the PyFR framework. The solver runs on a range of computer architectures, from laptops to the largest supercomputers, via a platform-unified templating approach that can generate/compile CUDA, OpenCL and C/OpenMP code at runtime. The extensibility of the cross-platform templating framework defined within PyFR is clearly demonstrated, as is the utility of P-multigrid for convergence acceleration. The platform independence of the solver is verified on Nvidia Tesla P100 GPUs and Intel Xeon Phi 7210 KNL manycore processors with a 3D Taylor-Green vortex test case. Additionally, the solver is applied to a 3D turbulent jet test case at Re = 10,000, and strong scaling is reported up to 144 GPUs. The new software constitutes the first high-order accurate cross-platform implementation of an incompressible Navier-Stokes solver via artificial compressibility and P-multigrid accelerated dual time stepping to be published in the literature. The technology has applications in a range of sectors, including the maritime and automotive industries. Moreover, due to its cross-platform nature, the technology is well placed to remain relevant in an era of rapidly evolving hardware architectures. Solution method: Artificial compressibility formulation discretised with a high-order Flux Reconstruction approach in space and P-multigrid accelerated dual time stepping in time. Program summary

An Artificial Compressibility Method for 1D Simulation of Open-Channel and Pressurized-Pipe Flow

“…More than 200 papers were retrieved by database searches for 2015-2020 using the terms artificial compressibility, pseudo compressibility, and dual time stepping. These studies include analysis of the relationships between AC and fractional step methods [86,87], studies of turbulent flow [88][89][90], development of discontinuous Galerkin methods [91][92][93], parallel solution of large-scale problems [94][95][96], multi-fluid simulations [97], further advances to entropically-damped AC methods [98][99][100], implementation high-order numerical schemes [101][102][103], use of characteristic methods [104][105][106], application for non-hydrostatic effects [107,108], magneto-hydrodynamic simulations [109][110][111], solution with lattice Boltzmann methods [112], and solution by smoothed particle hydrodynamics [113,114]. Note the above are examples and should not be considered an exhaustive list of recent work.…”

Non-Modal Analysis of Multigrid Schemes for the High-Order Flux Reconstruction Method

“…A drawback is that these methods are less mature and robust than their FVM counterparts. This has been the motivating factor in studying the numerical properties and stability limits of such schemes for under-resolved simulations of turbulent flows [4,5].…”

“…The Flux Reconstruction (FR) method, recently devised by Huynh [2,6] to solve conservation laws, is formulated as a unifying approach for discontinuous Finite Element methods such as DG and Spectral Difference (SD). The method has been further extended by Vincent, Castonguay and Jameson [7,8] through the family of Energy-Stable Flux Reconstruction (ESFR) schemes, and it has revealed itself particularly suited for the simulation of turbulent flows using implicit Large Eddy Simulations [4,5,9,10]. Several studies about the numerical properties of FR by means of von Neumann analysis exist in the literature [11,12,13,14], mainly in the context of linear advection and advection-diffusion.…”

Analysis of artificial pressure equations in numerical simulations of a turbulent channel flow

“…In that case, the system is for each physical timestep iterated in pseudo-time until convergence. This approach has been followed by various authors [42,31,48,39,1,41,36]. A review on the error analysis of artificial compressibility methods is given by Shen [57].…”

“…The lattice-Boltzmann (LB) method, which solves the Boltzmann transport equation on a discretised phase space [6], is also closely related to the AC method [28,53,5]. All these methods are explicit in time and local in space and thus particularly amenable to massively parallel GPU-based simulations and have low memory requirements [24,25,26,29,41].…”