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

University Of South Carolina At Columbia, Columbia SC

Investigators

Abstract

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.