A discrete variational derivative method for the Cahn–Hilliard equation with high-order spatial accuracy

We consider finite difference schemes for the Cahn–Hilliard equation, which describes spinodal decomposition. For this equation, stable numerical schemes can be designed by the discrete variational derivative method, but their spatial accuracy remained second-order, and it was difficult to improve under the standard boundary conditions. In this letter, we show that the discrete variational derivative method can be extended via the idea of summation-by-parts operators, by which we can construct spatially high-order schemes retaining the desired dissipation property. We also show numerical experiments to confirm the higher-order accuracy.