A Frozen-Mobility IMEXRK Framework for Variable-Mobility Gradient Flows

We propose and analyze a second-order accurate, predictor-corrector implicit-explicit Runge–Kutta (IMEXRK) framework for Allen–Cahn and Cahn–Hilliard equations with variable mobility. A first-order predictor yields an intermediate state, after which a two-stage, second-order IMEXRK corrector advances the solution while evaluating the mobility function explicitly and identically using the intermediate state in both stages. Thanks to the “frozen mobility” and stabilization terms, the original discrete energy dissipation law is established under an a priori maximum bound. However, justifying this bound is nontrivial, since the energy functional lacks H2 control and prevents direct Sobolev embedding. For constant mobility, we use coarse-refined estimates to derive global-in-time discrete H1 and H2 bounds under a mild time-step restriction τ≲ε6|lnε|−4, while a log-interpolation inequality gives the required maximum bound of O(ε−1|lnε|). The stabilization parameter is thus characterized as O(ε−2|lnε|2), without relying on global Lipschitz assumptions, a priori maximum bounds, or mesh-dependent time-step constraints. Numerical tests verify convergence and energy dissipation for variable-mobility gradient flows.