Benchmark Problems for the Numerical Schemes of the Phase‐Field Equations

Abstract

In this study, we present benchmark problems for the numerical methods of the phase-field equations. To find appropriate benchmark problems, we first perform a linear stability analysis and then take a growth mode solution as the benchmark problem, which is closely related to the dynamics of the original governing equations. As concrete examples, we perform convergence tests of the numerical methods of the Allen–Cahn (AC) and Cahn–Hilliard (CH) equations using the proposed benchmark problems. The one- and two-dimensional computational experiments confirm the accuracy and efficiency of the proposed scheme as the benchmark problems.

Integrating factor‐based time integrators for the Cahn–Hilliard equation

“…We have set the initial condition to u(x, 0) = cos(2x) + 0.01e cos(x+0.1) [38] for x ∈ (0, 2𝜋) and applied our proposed schemes to this initial condition until T = 2. The resulting plots can be seen in Figures 10 and 11.…”

“…To conduct stability tests on maximum admissible time step size for the proposed schemes, we set the initial condition to u(x, 0) = cos(2x) + 0.01e cos(x+0.1) [38] for x ∈ (0, 2𝜋). In Table 1, we list the maximum time step Δt max for different schemes by varying problem parameter 𝜖 and fixing the final time T = 2.…”

“…We have set the initial condition to $$$ u\left(x,0\right)=\cos (2x)+0.01{e}^{\cos \left(x+0.1\right)} $$$ [38] for $$$ x\in \left(0,2\pi \right) $$$ and applied our proposed schemes to this initial condition until $$$ T=2 $$$. The resulting plots can be seen in Figures 10 and 11.…”

“…To conduct stability tests on maximum admissible time step size for the proposed schemes, we set the initial condition to $$$ u\left(x,0\right)=\cos (2x)+0.01{e}^{\cos \left(x+0.1\right)} $$$ [38] for $$$ x\in \left(0,2\pi \right) $$$. In Table 1, we list the maximum time step $$$ \Delta {t}_{\mathrm{max}} $$$ for different schemes by varying problem parameter $$$ \epsilon $$$ and fixing the final time $$$ T=2 $$$.…”

A Novel Second-Order and Unconditionally Energy Stable Numerical Scheme for Allen–Cahn Equation

“…Nowadays, many numerical methods are applied to solve AC equations. Hwang et al [10] presented benchmark problems for the numerical methods of the phase eld equations.…”