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

一些抛物方程的快速稳定的基于紧指数时差的方法

• 批准号: 1521965
• 负责人: Lili Ju
• 金额: $ 20.1万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1521965&HistoricalAwards=false
• 关键词: Fast; Stable; Compact; Exponential; Time

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.

该项目的目标是开发和分析快速、稳定和准确的方法，用于科学和工程中不同应用中出现的一系列抛物线方程的数值解。该研究将为相变建模、化学反应、群体动力学、细胞膜建模、分子束外延、流体动力学和光传播等典型问题产生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。该项目还通过广泛的数值模拟，为理解微观结构粗化、膜脂囊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。尽管文献中已经广泛研究了基于指数时间积分器的技术来求解不同阶的半线性或非线性抛物型方程，但仍然缺乏对刚性非线性的精确和稳定处理、直接和显式结合各种非齐次边界条件的仔细的数值和理论研究，以及相应的快速实现算法。该项目中的方法本质上是明确的，它们将利用高阶有限差分的紧凑表示或矩形域中空间算子的谱近似、指数多步或龙格-库塔近似来实现边界和刚性非线性项的精确时间积分，有效增强数值稳定性的线性分裂方案，以及大大降低计算成本的基于FFT的快速计算。该研究将系统地研究几种提高紧凑指数时间差分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。 该项目还将推广并应用这些方法到一些生物和物理现象研究中产生的一些重要问题，例如细胞膜形状转变的相场弯曲能量模型和薄膜生长的分子束外延模型。