Energy-Dissipative Implicit-Explicit Runge-Kutta Schemes With an Optimized Stabilization Parameter for Directed Self-Assembly of Diblock Copolymer Melts

Abstract

We investigate high-order, energy-dissipative schemes for simulating the directed self-assembly (DSA) of diblock copolymer melts, which plays a crucial role in materials science, nanotechnology, and soft matter. The DSA process is primarily modulated through strategies such as external physical fields, substrate interfacial engineering, and geometric confinement, all of which are fundamentally described by Ohta–Kawasaki energy functionals. To preserve the intrinsic energy dissipation property of the corresponding gradient flow equations, we develop implicit-explicit Runge-Kutta (IMEXRK) schemes of up to third order and approximately fourth order, ensuring energy stability for any time step size. Under the uniform boundedness assumption of solutions, a novel criterion to justify the energy stability is established by rewriting the IMEXRK schemes in a unified matrix-vector framework. To address the time delay effect in stabilization schemes, an optimized, time-step-dependent selection of stabilization parameters is proposed, which shows significant accuracy improvement compared to a constant stabilization parameter. Numerical experiments validate the superior accuracy and stability, while simulations of directed self-assembly of diblock copolymer melts under different situations demonstrate the universality and broad applicability of the proposed schemes.