Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

We devise second-order accurate, unconditionally uniquely solvable and unconditionally energy stable schemes for the nonlocal Cahn–Hilliard (nCH) and nonlocal Allen–Cahn (nAC) equations for a large class of interaction kernels. We present numerical evidence that both schemes are convergent. We solve the nonlinear equations resulting from discretization using an efficient nonlinear multigrid method and demonstrate the performance of our algorithms by simulating nucleation and crystal growth for several different choices of interaction kernels.