## Unconditonally stable time steps for diffuse interface equations

### March 17, 2007

Authors: Benjamin P. Vollmayr-Lee and Andrew D. Rutenberg

Abstract:

We present Cahn-Hilliard and Allen-Cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy-controlled simulation. Our stability analysis is based on Eyre’s theorem and unconditional von Neumann stability analysis, both of which we present. Numerical tests confirm the accuracy of the von Neumann approach, which is straightforward and should be widely applicable in phase-field modeling. For the Cahn-Hilliard case, we show that accuracy can be controlled with an unbounded time step $\Delta t$ that grows with time $t$ as $\Delta t \sim t^{\alpha}$. We develop a classification scheme for the step exponent $\alpha$ and demonstrate that a class of simple linear algorithms gives $\alpha = 1/3$. For this class the speedup relative to a fixed time step grows with $N$, the linear size of the system, as $N/\ln N$. With conservative choices for the parameters controlling accuracy and finite-size effects we find that an $8192^{2}$ lattice can be integrated 300 times faster than with the Euler method.