A nonstiff, adaptive mesh refinement-based method for the Cahn–Hilliard equation

We present a nonstiff, fully adaptive mesh refinement-based method for the Cahn–Hilliard equation. The method is based on a semi-implicit splitting, in which linear leading order terms are extracted and discretized implicitly, combined with a robust adaptive spatial discretization. The fully discretized equation is written as a system which is efficiently solved on composite adaptive grids using the linear multigrid method without any constraint on the time step size. We demonstrate the efficacy of the method with numerical examples. Both the transient stage and the steady state solutions of spinodal decompositions are captured accurately with the proposed adaptive strategy. Employing this approach, we also identify several stationary solutions of that decomposition on the 2D torus.