A fourth-order spatial accurate and practically stable compact scheme for the Cahn-Hilliard equation

We present a fourth-order spatial accurate and practically stable compact difference scheme for the Cahn-Hilliard equation. The compact scheme is derived by combining a compact nine-point formula and linearly stabilized splitting scheme. The resulting system of discrete equations is solved by a multigrid method. Numerical experiments are conducted to verify the practical stability and fourth-order accuracy of the proposed scheme. We also demonstrate that the compact scheme is more robust and efficient than the non-compact fourth-order scheme by applying to parallel computing and adaptive mesh refinement.