A benchmark problem for the two- and three-dimensional Cahn-Hilliard equations

This paper proposes a benchmark problem for the two- and three-dimensional Cahn-Hilliard (CH) equations, which describe the process of phase separation. The CH equation is highly nonlinear and an analytical solution does not exist except trivial solutions. Therefore, we have to approximate the CH equation numerically. To test the accuracy of a numerical scheme, we have to resort to convergence tests, which consist of consecutive relative errors or a very fine solution from the numerical scheme. For a fair convergence test, we provide benchmark problems which are of the shrinking annulus and spherical shell type. We show numerical results by using the explicit Euler's scheme with a very fine time step size and also present a comparison test with Eyre's convex splitting schemes.