A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

Abstract

We present a very simple benchmark problem for the numerical methods of the Cahn–Hilliard (CH) equation. For the benchmark problem, we consider a cosine function as the initial condition. The periodic sinusoidal profile satisfies both the homogeneous and periodic boundary conditions. The strength of the proposed problem is that it is simpler than the previous works. For the benchmark numerical solution of the CH equation, we use a fourth-order Runge–Kutta method (RK4) for the temporal integration and a centered finite difference scheme for the spatial differential operator. Using the proposed benchmark problem solution, we perform the convergence tests for an unconditionally gradient stable scheme via linear convex splitting proposed by Eyre and the Crank–Nicolson scheme. We obtain the expected convergence rates in time for the numerical schemes for the one-, two-, and three-dimensional CH equations.

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

“…Wu et al [14] presented two benchmark problems on homogeneous and heterogeneous nucleation. Li et al [15] suggested the numerical benchmark solution of the CH equation. ey adopted the fourth-order Runge-Kutta method and finite difference method for the integration in time and the spatial differential operator, respectively, with a cosine initial condition.…”