<p>In this paper, we present and analyze a third-order (in time) numerical scheme for a binary fluid-surfactant phase field model which consists of two coupled Cahn-Hilliard type equations. The proposed scheme inherits the structure-preserving property of the model, including energy dissipation and bound preservation. To maintain the dissipation with respect to the original free energy, a linear convex splitting of the energy functional is first introduced. The numerical scheme is then developed by employing a spectral collocation approximation for spatial discretization and a third-order exponential time differencing Runge-Kutta scheme for time integration. Benefiting from the explicit evaluations of the nonlinear and coupling terms, the proposed scheme is linear and completely decoupled. By treating the numerical solution as the sum of the exact solution and a small perturbation, the bound-preserving property of the proposed scheme is proved along with its convergence. The original energy dissipation is further achieved with the help of the convex splitting method and the control of the nonlinear and coupling terms. Some numerical experiments in two and three dimensions are performed to verify the theoretical results and demonstrate the efficiency of the proposed scheme for simulations of phase separation phenomena.</p>

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A Third-Order Structure-Preserving Exponential Time Differencing Runge-Kutta Scheme for the Binary Fluid-Surfactant Phase Field Model

  • Jiayi Duan,
  • Lili Ju,
  • Xiao Li,
  • Zhonghua Qiao

摘要

In this paper, we present and analyze a third-order (in time) numerical scheme for a binary fluid-surfactant phase field model which consists of two coupled Cahn-Hilliard type equations. The proposed scheme inherits the structure-preserving property of the model, including energy dissipation and bound preservation. To maintain the dissipation with respect to the original free energy, a linear convex splitting of the energy functional is first introduced. The numerical scheme is then developed by employing a spectral collocation approximation for spatial discretization and a third-order exponential time differencing Runge-Kutta scheme for time integration. Benefiting from the explicit evaluations of the nonlinear and coupling terms, the proposed scheme is linear and completely decoupled. By treating the numerical solution as the sum of the exact solution and a small perturbation, the bound-preserving property of the proposed scheme is proved along with its convergence. The original energy dissipation is further achieved with the help of the convex splitting method and the control of the nonlinear and coupling terms. Some numerical experiments in two and three dimensions are performed to verify the theoretical results and demonstrate the efficiency of the proposed scheme for simulations of phase separation phenomena.