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

该项目的目标是开发和分析快速、稳定和准确的方法，用于在科学和工程中出现的各种应用的抛物方程的数值解。这项研究将为相变建模、化学反应、种群动力学、细胞膜建模、分子束外延、流体动力学和光传播等问题提供非常高效和有效的计算工具。精心设计的鲁棒高阶算法将使研究人员能够准确地捕捉这些系统的动态，而无需高昂的计算成本。本项目还通过广泛的数值模拟，为理解薄膜微观结构粗化、膜脂囊泡形状转变和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。尽管基于指数时间积分器的技术在求解不同阶的半线性或非线性抛物方程方面得到了广泛的研究，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，仍然缺乏细致的数值和理论研究。本项目中的方法本质上是显式的，它们将利用矩形域空间算子的高阶有限差分或谱近似的紧凑表示，指数多步或龙格-库塔近似用于边界和刚性非线性项的精确时间积分，线性分裂方案用于有效增强数值稳定性，以及基于fft的快速计算以大大降低计算成本。本研究将系统地研究几种提高紧凑指数时差方法在空间和时间上的精度和效率的技术，并对这些方案进行能量稳定性和误差分析。本项目还将这些方法推广并应用于一些生物和物理现象研究中出现的一些重要问题，如细胞膜形状转化的相场弯曲能模型和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.