Fully discrete Spectral-Galerkin scheme for a ternary Allen–Cahn type mass-conserved Nakazawa–Ohta phase-field model for triblock copolymers

Abstract

In this paper, we first use the Allen–Cahn relaxation dynamics to simulate the phase-field model of triblock copolymers, where the mass-conserving property is achieved by adding two additional nonlocal Lagrange multipliers. The numerical approximations of the obtained model are further considered, and an efficient fully-discrete numerical scheme based on the Spectral-Galerkin approach for spatial discretization and the so-called explicit-IEQ (invariant energy quadratization) method for time marching is developed. Two auxiliary variables are introduced to reformulate the governing system into an equivalent form, which facilitates the design of numerical algorithms by simply discretizing nonlinear terms explicitly. The resulting scheme is not only second-order accurate in time and spectrally accurate in space, but also easy to implement, i.e., the scheme can be carried out by only solving multiple independently decoupled, linear, and constant-coefficient elliptic equations at each time step. We also prove the unconditional energy stability of the developed scheme and demonstrate its effectiveness by implementing several benchmarking numerical examples in 2D and 3D.