Mesh-Robustness of an Energy Stable BDF2 Scheme with Variable Steps for the Cahn–Hilliard Model

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn–Hilliard model. We focus on numerical influences of time-step variations by using recent theoretical framework discrete orthogonal convolution kernels. Some novel embedding inequalities respect kernels are developed such that a concise $$L^2$$ norm error estimate established at first time under updated step-ratio restriction $$0 <r_k:=\tau _k/\tau _{k-1}\le r_{\mathrm {user}}$$ , where $$r_{\mathrm can be chosen user {user}}<4.864$$ . stabilized BDF2 shown mesh-robustly convergent in sense convergence constant (prefactor) independent adjoint ratios. suggested method proved preserve modified dissipation law levels if $$0<r_k\le it norm. On basis ample tests random meshes, useful adaptive time-stepping strategy efficiently capture multi-scale behaviors and accelerate long-time simulation approaching steady state.