论文标题
具有可变界面参数的Cahn-Hilliard方程的阳性二阶BDF方案
A positivity-preserving second-order BDF scheme for the Cahn-Hilliard equation with variable interfacial parameters
论文作者
论文摘要
我们为大分子微球复合水凝胶，时间依赖性的金茨堡 - 兰道（MMC-TDGL）方程提供了新的二阶有限差差方案，这是一种具有Flory-Huggins-Degennes能量电位的Cahn-Hilliard方程。这种具有无条件能量稳定性的数值方案基于向后分化公式（BDF）方法派生与Douglas-Dupont正则化项相结合的时间推导。此外，我们在理论层面上为提出的方案提供了数值解决方案的点结合。为了进行收敛分析，我们将三个非线性对数项整体视为，并通过使用非线性误差内部产品始终非负值的属性直接处理所有对数项。此外，我们在$ \ ell^\ infty（0，t; h_h^{ - 1}）\ cap \ ell^2（0，t; h_h^1）$ norm中介绍了详细的收敛分析。最后，我们使用局部牛顿近似和多机方法来求解非线性数值方案，并提供了各种数值结果，包括数值收敛测试，阳性性能属性测试，旋转性分解，能量耗散，耗散和质量保护特性。
We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method time derivation combining with Douglas-Dupont regularization term. In addition, we present a point-wise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.
