Linear energy-stable method with correction technique for the Ohta–Kawasaki–Navier–Stokes model of incompressible diblock copolymer melt

Abstract

An efficiently linear time-marching method is proposed for the incompressible fluid flows coupled Ohta–Kawasaki model of diblock copolymer melt. Although this model satisfies the energy dissipation law under appropriate boundary conditions, the existences of nonlinear terms and coupling terms lead to difficulties of designing energy-stable numerical algorithms. To overcome the difficulties resulting from nonlinear and coupling terms, we herein construct a numerical method based on a highly efficient variant of scalar auxiliary variable approach. The stabilization technique is introduced to maintain the stability with large time steps. To improve the consistency, a simple and practical relaxation strategy is used to correct the difference between time-discretized modified energy and original energy. In each time step, all variables are totally decoupled and we only need to solve linear elliptic equations in a step-by-step manner. The unique solvability and energy stability are analytically proved. Extensive numerical experiments indicate that the proposed method not only achieves desired accuracy, energy stability, and consistency, but also has good capabilities on simulating various hydrodynamically coupled pattern formations.