Collaborative Research: Efficient, Accurate, and Structure-Preserving Numerical Methods for Phase Fields-Type Models with Applications

该项目将开发在原子和纳米尺度上模拟材料相变的计算方法，旨在了解大时间尺度上的行为。通过这些模拟，该项目将有助于理解复杂的生物生长和癌症、与生物生长和发育相关的多相活性粒子和离子流体等过程，以及研究物理学和材料工程中的其他复杂现象。重点将放在一类特殊的模型上：具有奇异能量势的梯度流方程。该项目将开发理论和软件；该项目中开发的代码将被放大以进行真实世界的三维模拟。此外，一些有待开发的数值算法可能会对深度学习领域产生影响。该项目将为两所相关机构的研究生和本科生提供跨学科的应用数学和科学计算培训和研究经验。在所提出的梯度流模型中，能量势涉及奇异性，因此正性保持性质成为使数值逼近更好地定义的关键特征。此外，对于这些具有奇异能量势的梯度模型，如描述表面扩散的双简并Cahn-Hilliard模型、模拟凝固过程的具有热输运的新相场晶体模型、用于两相密度失配流动的新的准不可压缩Cahn-Hilliard-Navier-Stokes模型、用于离子混合物的Poisson-Nernst-Plank模型以及多相磁流体动力学方程，将考虑能量稳定性和最优率收敛分析。将使用新的有限差分、混合有限元和/或傅里叶伪谱空间近似。详细研究了三阶时间精度的收敛分析，这将是第一次对具有奇异势的梯度流进行这种分析。此外，还将设计和分析这些高度非线性格式的数值解算器，基于预条件最陡下降和Nester ov加速方法。基于高效的自适应非线性多重网格方法也将得到详细的测试和研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project will develop computational methods for simulations of phase transformations in materials at the atomic and nanometer scales, aiming at understanding behavior at large time scales. With these simulations, the project will contribute to the understanding of processes such as complex biological growth and cancer, multi-phase active-particle and ionic fluids relevant in biological growth and development, and in the study of other complex phenomena in physics, and material engineering. The focus will be on a particular class of models: gradient flow equations with singular energy potentials. The project will develop theory and software; the codes developed in this project will be scaled up to conduct real-world three-dimensional simulations. In addition, some numerical algorithms to be developed could impact the field of deep learning. This project will provide interdisciplinary applied mathematics and scientific computing training and research experiences for both graduate and undergraduate students at the two institutions involved.In the proposed gradient flow models, a singularity is involved in the energy potential, so that the positivity-preserving property becomes a crucial feature to make the numerical approximation well-defined. In addition, energy stability and optimal rate convergence analysis will be considered for these gradient model with singular energy potential, such as the doubly degenerate Cahn-Hilliard model describing surface diffusion, a new phase field crystal model with heat transport for simulating solidification, a new quasi-incompressible Cahn-Hilliard-Navier-Stokes model for two-phase density mismatched flow, the Poisson-Nernst-Plank model for ionic mixtures, and multi-phase magneto-hydrodynamics equations. Novel finite difference, mixed finite element, and/or Fourier pseudo-spectral spatial approximations will be utilized. Convergence analysis up to the third order temporal accuracy will be investigated in details, which will be the first such work for gradient flows with singular potential. Moreover, numerical solvers for these highly nonlinear schemes will be designed and analyzed, based on the preconditioned steepest decent and Nesterov accelerated methods. Highly efficient adaptive nonlinear multigrid methods based will also be tested and studied in details.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.