Design and Analysis of Highly Efficient Algorithms for Complex Nonlinear Systems

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This project focuses on the development and analysis of innovative, structure preserving algorithms for complex nonlinear systems in science and engineering applications. This project will not only lead to efficient numerical algorithms for a large class of complex nonlinear systems of current interests, but also contribute through numerical simulation to a better understanding of some fundamental issues in materials science, fluid mechanics and other related fields. This project will also provide opportunities for the involved students to learn critical skills of computational and applied mathematics and to develop state-of-the-art numerical algorithms for science and engineering applications. This project will support one graduate student per year.Complex nonlinear systems that possess dissipative or conservation properties are ubiquitous in modeling of real-world phenomena. It is a major challenge to construct efficient and accurate numerical schemes that can preserve important dissipative or conservation properties, and in certain cases, positivity of physical variables. This project will overcome these challenges by extending the flexible and robust scalar auxiliary variable or SAV approach, which has proven to be highly effective for gradient flows. In particular, the SAV approach will be extended to deal with additional difficulties such as those in (i) systems with physical constraints such as mass and/or surface area conservations; (ii) highly anisotropic systems and systems with nonlinear mobilities; (iii) systems with positivity preserving or maximum principles; (iv) systems coupling gradient flows with other conservation laws, and (v) optimizations. The proposed methodology will lead to numerical predictive tools that extend the applicability of mathematical and experimental analysis, and contribute to better understanding of complex physical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.