We　perform　a　comparison　study　on　the　different　dynamics　between　the　Allen–Cahn　(AC)　and　the　Cahn–Hilliard　(CH)　equations.　The　AC　equation　describes　the　evolution　of　a　non-conserved　order　field　during　anti-phase　domain　coarsening.　The　CH　equation　describes　the　process　of　phase　separation　of　a　conserved　order　field.　The　AC　and　the　CH　equations　are　second-order　and　fourth-order　nonlinear　parabolic　partial　differential　equations,　respectively.　Linear　stability　analysis　shows　that　growing　and　decaying　modes　for　both　the　equations　are　the　same.　While　the　growth　rates　are　monotonically　decreasing　with　respect　to　the　modes　for　the　AC　equation,　the　growth　rates　for　the　CH　equation　are　increasing　and　then　decreasing　with　respect　to　the　modes.　We　perform　various　numerical　tests　using　the　Fourier　spectral　method　to　highlight　the　different　evolutionary　dynamics　between　the　AC　and　the　CH　equations.　©　2018　Elsevier　Ltd