Collaborative Research: Efficient, Stable and Accurate Numerical Algorithms for a class of Gradient Flow Systems and their Applications

Purdue University, West Lafayette IN

Investigators

Abstract

This project focuses on the development of efficient, stable and accurate numerical algorithms for gradient flow systems which are ubiquitous in modeling of real-world phenomena. It is expected that the research will not only lead to efficient numerical algorithms for a class of problems of current interests, but also contribute to better understandings of some fundamental issues in materials science, biotechnology, and other related fields through numerical simulations. This project will also provide opportunities for the involved graduate and undergraduate students to learn critical skills of computational and applied mathematics and to develop state-of-the-art numerical algorithms for science and engineering applications. The free energies of gradient flow systems usually consist of various nonlinear potentials formulated in diverse complex formats which present a major challenge in the construction of efficient and accurate time discretization schemes. The project aims at overcoming this challenge by using a flexible and robust IEQ approach that enables one to develop time discretization schemes for a large class of gradient flow systems. The goals of this proposal are three folds: (i) to develop a unified numerical framework of time-marching schemes for solving general gradient flow models with high nonlinearity; (ii) to develop efficient numerical schemes for a number of challenging gradient flow models of current interests (e.g., nonlinear coupled multivariable models, nonlocal models, anisotropic models, nonlinear coupled systems that follow various physical principles, tensor based liquid crystal models); (iii) to investigate some fundamental issues of viscoelastic drops on substrates and active liquid crystal droplets using the developed predictive numerical tools. The proposed schemes will lead to numerical predictive tools that extend the capability of mathematical and experimental analysis, and contribute to better understanding of some pressing science and engineering issues related to multi-phase complex fluids.