液体薄膜粗化模型的保结构、能量稳定数值格式
Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model
摘要：
本文针对液滴液膜模型提出了两种有限差分格式，并分析了它们的优缺点。一阶和二阶精确的时间算法被认为是。在一阶格式中，凸势项和表面扩散项是隐式的，而凹势项是显式更新的。此外，我们提供了一个理论上的理由，这种数值算法有一个唯一的解决方案，这样的积极性总是保持在逐点水平的相位变量，从而避免了奇异性的计划。事实上，Leonard-Jones势项在0值附近的奇异性阻止了数值解达到这样的奇异值，使得正性结构总是被保持。此外，在对时间步长没有任何限制的情况下，得到了该数值格式的无条件能量稳定性。在二阶数值格式中，应用BDF时间模板，并推导出交替的凹凸分解，使得凹部对应于二次能量。相应地，隐式地处理了组合Leonard-Jones势项，用二阶Adams-Bashforth显式外推近似了凹部，并加入了人工Douglas-Dupont正则化项以保证能量稳定性.二阶格式的唯一可解性和保正性可以类似地建立。此外，对一阶和二阶精度格式的最优速度收敛性进行了分析。最后给出了几个数值模拟结果，证明了数值格式的鲁棒性。
英文摘要：
In this paper, two finite difference numerical schemes are proposed and analyzed for the droplet liquid film model, with a singular Leonard-Jones energy potential involved. Both first and second order accurate temporal algorithms are considered. In the first order scheme, the convex potential and the surface diffusion terms are implicitly, while the concave potential term is updated explicitly. Furthermore, we provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity is always preserved for the phase variable at a point-wise level, so that a singularity is avoided in the scheme. In fact, the singular nature of the Leonard-Jones potential term around the value of 0 prevents the numerical solution reaching such singular value, so that the positivity structure is always preserved. Moreover, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. In the second order numerical scheme, the BDF temporal stencil is applied, and an alternate convex-concave decomposition is derived, so that the concave part corresponds to a quadratic energy. In turn, the combined Leonard-Jones potential term is treated implicitly, and the concave part the is approximated by a second order Adams-Bashforth explicit extrapolation, and an artificial Douglas-Dupont regularization term is added to ensure the energy stability. The unique solvability and the positivity-preserving property for the second order scheme could be similarly established. In addition, optimal rate convergence analysis is provided for both the first and second order accurate schemes. A few numerical simulation results are also presented, which demonstrate the robustness of the numerical schemes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
