Fast and Stable Compact Exponential Time Difference Based Methods for Some Parabolic Equations

这个项目的目标是开发和分析一类抛物型方程的快速、稳定和准确的数值解方法，这些方程在科学和工程中有不同的应用。这项研究将产生非常有效的计算工具，用于解决以相变建模、化学反应、布居动力学、细胞膜建模、分子束外延、流体动力学和光传播为代表的问题。设计良好的稳健高阶算法将使研究人员能够在不增加计算成本的情况下准确地捕捉到这些系统的动态。该项目还通过广泛的数值模拟，为理解薄膜微结构粗化、膜脂泡形状转变和薄膜外延生长的动力学过程提供了新的见解。研究生将直接参与前沿研究并从中受益。虽然基于指数时间积分器的方法在求解不同阶的半线性或非线性抛物型方程方面已经得到了广泛的研究，但对于刚性非线性的精确和稳定处理、各种非齐次边界条件的直接和显式结合以及相应的快速实现算法，仍然缺乏认真的数值和理论研究。该项目中的方法本质上是显式的，它们将使用矩形区域中空间算子的高阶有限差分或谱近似的紧致表示，用于边界和刚性非线性项的精确时间积分的指数多步或龙格-库塔近似，用于有效提高数值稳定性的线性分裂格式，以及用于极大地降低计算成本的基于FFT的快速计算。该研究将系统地研究提高紧致指数时差方法在空间和时间上的精度和效率的几种技术，并对这些方案进行能量稳定性分析和误差分析。该项目还将这些方法推广和应用于一些生物和物理现象研究中的一些重要问题，如用于细胞膜形状转变的相场弯曲能模型和用于薄膜生长的分子束外延模型。

The goal of this project is to develop and analyze fast, stable, and accurate methods for numerical solutions of a family of parabolic equations that appear in diverse applications in science and engineering. The research will lead to production of very efficient and effective computational tools for problems typified by phase transition modeling, chemical reactions, population dynamics, cell membrane modeling, molecular beam epitaxy, fluid dynamics, and light propagation. The well-designed robust high-order algorithms would allow researchers to accurately catch the dynamics of these systems without high computational costs. This project also offers new insights into the understanding of the kinetic processes of microstructure coarsening, shape transformation of membrane lipid vesicles, and epitaxial growth of thin films through extensive numerical simulations. Graduate students will be directly involved in and benefit from their participation in the frontier research. Although exponential time integrator based techniques have been widely researched in the literature for solving semilinear or nonlinear parabolic equations of different orders, there still lack careful numerical and theoretical studies on accurate and stable treatments of stiff nonlinearities, direct and explicit incorporation of various inhomogeneous boundary conditions, and corresponding fast implementation algorithms. The methods in this project are explicit in nature, and they will utilize compact representations of high-order finite differences or spectral approximations for spatial operators in a rectangular domain, exponential multistep or Runge-Kutta approximations for accurate time integrations of boundary and stiff nonlinear terms, linear splitting schemes for effectively enhancing numerical stabilities, and FFT-based fast calculations for greatly reducing computational costs. The research will systematically study several techniques for improving accuracy and efficiency of the compact exponential time differencing methods in both space and time, and develop energy stability and error analyses for these schemes. The project will also generalize and apply these methods to some important problems arising from the study of some biological and physical phenomena, such as phase field bending energy models for cell membrane shape transformation and molecular beam epitaxy models for thin film growth.