Maximum Bound Principle-Preserving Time Integration Methods for Some Semilinear Parabolic Equations

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The project will focus on the algorithms and analysis for a class of equations with applications in fluids dynamics, solid mechanics, materials science, chemistry, and cell biology to image and data sciences. Specifically, a class of semilinear parabolic equations will be considered, and the focus will be on ensuring that the computational solutions possess particular important properties called the time-invariant maximum bound principle (MBP). Such properties are important for the models of grain growth and coarsening, thin film microstructure, crystal growth, dislocation-solute interactions, and image restoration and deblurring. The project will design and analyze efficient and accurate MBP-preserving time integration methods of high-order accuracy. The project will develop and disseminate software and provide an interdisciplinary training opportunity for graduate students.The proposed activities contain diverse research topics in computational and applied mathematics, ranging from algorithm design, numerical analysis, and efficient implementation to practical applications in science and engineering. Specifically, the project presents an important step toward developing and analyzing efficient and high-order accurate MBP-preserving time integration methods. Rigorous analysis for a wide class of semilinear parabolic equations within or beyond the current analytical framework will be carried out. This project will not only lead to significant innovations in numerical tools and computer codes for solving these types of equations, but also offer new insights into a number of outstanding theoretical issues on MBP preservation and energy stability in both time-continuous and time-discrete settings. The goals include the design and analysis of linear schemes with high-order accuracy based on the Runge-Kutta integrating factor and the modified scalar auxiliary variable approaches. The project will also extend and develop MBP-preserving time integration methods for some important phase field models beyond the existing analytical framework, including but not limited to, the mass-conserving Allen-Cahn equations with different types of constraints, the convective Allen-Cahn equation and the coupled Navier-Stokes/Allen-Cahn system. These research problems are very useful and challenging with important applications in science and engineering.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.