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，稳定和精确的方法，用于科学和工程中各种应用中出现的抛物方程族的数值解。该研究将导致生产非常高效和有效的计算工具，用于相变建模，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典型问题。设计良好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，而无需高计算成本。该项目还提供了新的见解，通过广泛的数值模拟的微观结构粗化，膜脂囊泡的形状转变，和薄膜的外延生长的动力学过程的理解。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技术已经在文献中被广泛研究用于求解不同阶的半线性或非线性抛物型方程，但是仍然缺乏对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合并以及相应的快速实现算法的仔细的数值和理论研究。在这个项目中的方法是明确的性质，他们将利用紧凑表示的高阶有限差分或频谱近似的空间运营商在一个矩形域，指数多步或龙格库塔近似的精确时间积分的边界和刚性非线性项，线性分裂计划，有效地提高数值稳定性，和基于FFT的快速计算，大大降低计算成本。本研究将系统地研究在空间和时间上提高紧致指数时间差分方法精度和效率的几种技术，并对这些格式进行能量稳定性和误差分析。 本项目还将把这些方法推广应用于一些生物和物理现象研究中出现的重要问题，如细胞膜形状转变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.