Numerical Approximation of the Nonlocal Ternary Viscous Cahn–Hilliard Model Based on a High Order Explicit Hybrid Algorithm
Zhifeng Weng, Shuying Zhai, Xinlong Feng
Source abstract
Abstract. We consider a nonlocal ternary viscous Cahn–Hilliard (tVCH) model, which incorporates a finite range of spatial nonlocal interactions that are essential for capturing sharp interfaces among bulk phases and for the accurate calculation of contact angles. The model is to encapsulate the nonlocal versions of three popular phase separation models: the ternary Cahn–Hilliard (CH) model, the constrained ternary Allen–Cahn (AC) model, and the ternary modified Cahn–Hilliard model. However, there are very few works providing a comparative study on dynamical behaviors of these three models; moreover, numerical studies for the nonlocal tVCH model are still lacking. It is therefore quite challenging to develop highly accurate, stable, and efficient numerical schemes for the nonlocal tVCH model. With the combination of splitting method and spectral deferred correction (SDC) method, we present a high-order explicit hybrid algorithm for solving the nonlocal tVCH model. We start at a second-order time-splitting scheme based on the Fourier spectral method and the strong stability preserving Runge–Kutta method, so that the scheme can avoid nonlinear iteration and reach spectral accuracy in space. The stability and convergence analyses of the scheme are then analyzed. To improve its temporal accuracy, a semi-implicit SDC scheme is further developed. Extensive numerical experiments, including accuracy comparisons with recently popular numerical methods, stability tests, and contact angle tests, are performed to validate the theoretical analysis and the efficiency of the present method. Results indicate that the nonlocal tVCH model is more suitable than the local and fractional-order versions for accurately calculating the contact angles among three phases. The viscoelastic creep phenomenon of nonlocal model in the numerical results could contribute to design high strength and tender fabric composite materials. The present method can be extended to solve the nonlocal tVCH with convection terms.
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