Maximally fast coarsening algorithms

We present maximally fast numerical algorithms for conserved coarsening systems that are stable and accurate with a growing natural time step $\Delta t=A{t}_s^{2∕3}$. We compare the scaling structure obtained from our maximally fast conserved systems directly against the standard fixed time-step Euler algorithm, and find that the error scales as $\sqrt{A}$—so arbitrary accuracy can be achieved. For nonconserved systems, only effectively finite time steps are accessible for similar unconditionally stable algorithms.

Figure 1

The average scaled structure $\tilde{S}\left(x\right)\equiv {ϵ}^{2}S\left(xϵ\right)$ vs $x\equiv k∕ϵ$ for $L_{\infty }=512$ system at $t_s=1024$, with the Euler update (circles) with $\Delta t=0.03$ and the Eyre update (“$+$”) with $\Delta t=A{t}_s^{2∕3}$, where $A=0.01$. Triangles indicate the absolute difference between Eyre and Euler updates, $\mid \Delta \tilde{S}\left(x\right)\mid$.

Figure 2

The Eyre structural error vs $A$ for Eyre algorithms driven at the natural time step $\Delta t=A{t}_s^{2∕3}$. The error is the maximum absolute difference between the Eyre and the Euler correlations. Shown is data for $L_{\infty }=256$ (open squares) and $L_{\infty }=512$ (open circles). Also shown as filled symbols are the asymptotic values of $1-\Delta t_s∕\Delta t$ for various $A$. Both errors exhibit $\sqrt{A}$ behavior, as discussed in the text.

Figure 3

For nonconserved update driven with $\Delta t=\infty$, we plot ${\widetilde{\Delta t}}_s\equiv \Delta t_s\sqrt{(a_1-1)(1-a_2)}$ vs $\tilde{a}\equiv \sqrt{(1-a_2)∕(a_1-1)}$ showing the prediction (solid line with $\xi =0.85$) of Eq. (20).