Accuracy of the lattice-Boltzmann method using the Cell processor

Accelerator processors like the new Cell processor are extending the traditional platforms for scientific computation, allowing orders of magnitude more floating-point operations per second (flops) compared to standard central processing units. However, they currently lack double-precision support and support for some IEEE 754 capabilities. In this work, we develop a lattice-Boltzmann (LB) code to run on the Cell processor and test the accuracy of this lattice method on this platform. We run tests for different flow topologies, boundary conditions, and Reynolds numbers in the range $\mathrm{Re}=6–350$. In one case, simulation results show a reduced mass and momentum conservation compared to an equivalent double-precision LB implementation. All other cases demonstrate the utility of the Cell processor for fluid dynamics simulations. Benchmarks on two Cell-based platforms are performed, the Sony Playstation3 and the QS20/QS21 IBM blade, obtaining a speed-up factor of 7 and 21, respectively, compared to the original PC version of the code, and a conservative sustained performance of 28 gigaflops per single Cell processor. Our results suggest that choice of IEEE 754 rounding mode is possibly as important as double-precision support for this specific scientific application.

Figure 1

(Color online) ${\mathrm{LB}}_{\mathrm{SPE}}$ percentage of mass (a) and momentum (b) lost against time for a relaxing random velocity field using different fluid viscosities $\nu =0.018,0.071,0.167,0.5$ on a $128\times 128$ lattice. Small but systematic mass or momentum loss is obtained using ${\mathrm{LB}}_{\mathrm{SPE}}$. Mass or momentum loss increases as fluid viscosity decreases. Continuous line with points shows a negligible mass or momentum increase using ${\mathrm{LB}}_{\mathrm{PPE}}$.

Figure 2

(Color online) $\mathbf{u}{\left(\mathbf{x}\right)}^2$ at $t=200000$ of flow past a cylinder. The slower the velocity, the darker the pixel. (a) The case of a viscous fluid $(\mathrm{Re}=12.8)$. (b) The beginning of vortex production $(\mathrm{Re}=355.5)$. (c) The velocity field profiles $u_x$ at $L_y∕2$ for two simulations of relaxing waves with $\nu =0.018$ (dashed lines) and $\nu =0.5$ at different simulation times. (d) The velocity profile $u_x$ against $y$ for a Poiseuille $\nu =0.5$ flow. Simulation results on the Cell processor and on a PC coincide. Also, the profiles obtained agree with the profile fit (dashed line) using Eq. (4).

Figure 3

(a) Percentage error $({\nu }_{\text{input}}-{\nu }_{\mathrm{eff}})∕{\nu }_{\text{input}}$ depending on input fluid viscosity ${\nu }_{\text{input}}$ when simulating wave relaxation in a $64\times 64$ box. Input and effective viscosities differ by less than 0.02%. (b) Percentage error $({\nu }_{\text{input}}-{\nu }_{\mathrm{eff}})∕{\nu }_{\text{input}}$ depending on input fluid viscosity ${\nu }_{\text{input}}$ obtained by simulating Poiseuille flow on a $64\times 32$ lattice. ${\nu }_{\mathrm{eff}}$ agrees well with ${\nu }_{\text{input}}$.