Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards

The octahedron model introduced recently has been implemented onto graphics cards, which permits extremely large-scale simulations via binary lattice gases and bit-coded algorithms. We confirm scaling behavior belonging to the two-dimensional Kardar-Parisi-Zhang universality class and find a surface growth exponent: $\beta =0.2415\left(15\right)$ on $2^{17}\times 2^{17}$ systems, ruling out $\beta =1/4$ suggested by field theory. The maximum speedup with respect to a single CPU is 240. The steady state has been analyzed by finite-size scaling and a growth exponent $\alpha =0.393\left(4\right)$ is found. Correction-to-scaling-exponent are computed and the power-spectrum density of the steady state is determined. We calculate the universal scaling functions and cumulants and show that the limit distribution can be obtained by the sizes considered. We provide numerical fitting for the small and large tail behavior of the steady-state scaling function of the interface width.

Figure 1

Sketch of the dead border domain decomposition scheme employed. Lattice sites are indicated by dots, where the gray dots represent inactive sites. Since only two slopes relating any sites to its nearest neighbors are stored off site only two edges are inactive.

Figure 2

Time in seconds for one MCS on a C2070 (black bullets, left axis). Speedup over a nonparallel implementation on a Core i5 750@3GHz (red squares, right axis). The jump in speedup between ${\log }_{2}L=12$ and 13 results from the system exceeding the CPUs L3 cache at this point. The maximum speedup factor achieved was roughly 240.

Figure 3

Local slopes of the surface growth for different sizes ($L=2^{12},2^{13},2^{16},2^{17}$). Averaging was done over 20–100 independent runs.

Figure 4

Local slopes of the roughness exponent for different sizes. The line shows a fit with the form (14). (Inset) Surface width of the stationary state as the function of linear system size. The line corresponds to a fit with the form (10).

Figure 5

Finite-size scaling of $W(L,t)$ for $L=2^{12},2^{13},2^{16},2^{17}$. (Right inset) PSD of the $L=2^{13}$ system in the steady state. (Left inset) Distribution of $W/t^{0.2415}$ in the growth phase of the $L=2^{17}$ system.

Figure 6

The universal scaling function ${\Psi }_L\left(x\right)$ in the steady state for $L=2^8,2^9,2^{10},2^{11},2^{12}$. The dotted line shows the $L=157$ results of Ref. [22].

Figure 7

FSS of the cumulants: ${\kappa }_1$, ${\kappa }_2$, ${\kappa }_3$, ${\kappa }_4$ (top to bottom) for $L=512,1024,2048,4096,8192$. The lines correspond to power-law fitting with very small exponents.

Figure 8

Small (left) and large (right) asymptotics of ${\Psi }_L\left(x\right)$. (Dotted blue line) $L=157$ of Ref. [42]; (red line) $L=2048$ data; (dashed line) fitting with the form (22). (Right) linear-logarithmic fitting (dashed line) to the large $x$ part of the same data.