大分子微球复合水凝胶(Macromolecular Microsphere Composite Hydrogel, 简称 MMC Hydrogel)是一种具有极高机械强度特性的新型水凝胶, 在生物化学与医学等领域具有重要的应用价值. 大分子微球复合水凝胶相变模型是借助经典 Flory-Huggins 晶格理论及依赖于时间的 Ginzburg-Landau 介观模拟方法而得到的用来刻画 MMC 水凝胶微观相结构变化的数学模型, 它是一个带奇性的且满足能量耗散的四阶抛物型偏微分方程或方程组. 这种模型在数值模拟方面已经有了一定的实验结果, 只是在理论上的研究工作还不多. 本文主要针对两个 MMC 水凝胶相变模型分别设计出保正且稳定的数值格式并对格式进行严格的理论分析, 最后通过数值模拟观察 MMC 水凝胶的微观相变过程. 本文研究的问题主要有以下四个创新点.

第一, 由于模型方程中含有带对数的项以及非线性扩散系数 κ(φ), 这两项都是高度非线性的且函数本身存在奇点, 这为理论分析带来了一定的困难. 本文充分利用对数能量自身的奇异性证明了所提格式的数值解的存在惟一性, 同时找到了关于 deGennes 扩散系数 κ(φ)的一个不等式, 其在证明数值解有界且其值域包含于 (0,1/ρ ) 的过程中起到了关键的作用. 另外由于凸分裂方法构造的非线性格式具有较好的分析性质, 因此我们所提出的格式从理论上可以得到无条件能量稳定性质, 这一性质是热力学一致的.

第二, 充分利用非线性能量与扩散项的凸性对格式进行严格的收敛性分析, 其中扩散项的一部分可以控制由凹项带来的坏项.

第三, 针对大分子微球复合水凝胶三组分相变系统, 构造了关于时间一阶精度的无条件能量稳定的有限差分格式, 并严格从理论上证明了该格式有且仅有一对数值解 (φ₁ ,φ₂ ), 并且在逐点意义下φ₁ ,φ₂ 与 φ₃ 不仅是有界的而且其值域均包含于 (0,1), 即保证对数运算有意义. 最后从数值算例上验证了所提出的数值格式的有效性及收敛性等性质, 例如离散情况下的能量耗散性, 质量守恒性, 数值解的有界性, 收敛阶及 MMC 水凝胶微观相结构的变化等. 通过数值实验进一步验证了理论的正确性.

第四, 本文对提到的三个数值格式均给出了严格的误差分析. 特别地, 针对具有非线性表面扩散系数的三组分 MMC 水凝胶相变系统, 在证明其收敛性分析时用到了一些非常规的技巧, 例如高阶渐近展开(达到二阶时间精度), 数值解的粗估计(用于确定数值解的上界)与细估计等.

Macromolecular microphere composite (MMC) hydrogel, a new kind of hydrogel, is widely used in the field of biomedicine due to its high mechanical strength. MMC phase transition model, based on the classical Flory-Huggins lattice theory and the time-dependent Ginzburg-Landau mesoscopic technique, is a new mathematic model. It describes the microscopic structure and the phase transition of the MMC hydrogel. It is a kind of fourth order parabolic partial differential equations with the singular nature of the logarithmic energy potential and satisfies energy dissipation. There have been existing many numerical experiment, but these are few research work at theoretical level. In this thesis, we not only design positivity-preserving and stable numerical schemes with first order and second order accuracy but also present the corresponding convergence analysis and perform simulations for two MMC phase mathematical models.

There are mainly four innovations in this thesis.

First, there are some challenges in the theory analysis due to the existence of the logarithmic term and nonlinear diffusive coefficient κ(φ) for the phase models. This thesis makes full use of the singularity of the energy density to prove the unique solvability of the supposed numerical schemes. In particular, the natural structure of the deGennes diffusive coefficient also ensures the desired positivity-preserving property. In turn, the unconditional energy stability becomes an outcome of the unique solvability and the convex-concave decomposition for the energy functional.

Second, making full use of the convexity of the nonlinear energy and diffusion terms, we present the convergence analysis, in which a part of diffusion term could control the bad result coming from the concave term.

Third, for ternary MMC system, based on a convex splitting of the energy functional, we propose an unconditionally energy stable difference scheme constructed with first order accuracy in time and second order accuracy in space, and the positivity-preserving property and unconditional energy stability are established.

Fourth, for three numerical schemes developed in this thesis, we present the corresponding convergence analysis. In particular, for the ternary MMC systems, we rigorously prove first order convergence in time and second order convergence in space for the numerical scheme. Many highly non-standard skills have to be involved, due to the nonlinear and singular nature of the surface diffusion coefficients. The higher order asymptotic expansion (up to second order temporal accuracy), the rough error estimate (to establish the ‘ ∞ bound for the phase variables), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, it will be the first work to provide an optimal rate convergence estimate for a ternary phase field system with singular energy potential.