Higher order spectral element scheme for two- and three-dimensional Cahn–Hilliard equation

Abstract

Higher order spectral element scheme is presented for Cahn–Hilliard equation in two and three dimensions. Legendre polynomial based nodal spectral element method is employed in space whereas explicit and semi-implicit schemes are discussed for time discretization. Cahn–Hilliard equation conserves mass whereas it dissipates energy with time. These two properties are used to test the robustness and accuracy of the proposed numerical scheme. Semi-implicit time stepping numerical scheme which is unconditionally gradient stable is used for the numerical simulation. Such schemes require that the Newton’s iterative solver which is used to solve the fully discretized nonlinear equations must converge to the correct solution. To ensure the convergence, the continuous version of Newton’s iterative solver is used. Various test cases are solved in two and three dimensions showing the efficacy of the proposed scheme.