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

该项目将专注于一类方程的算法和分析，并应用于流体动力学，固体力学，材料科学，化学和细胞生物学以及图像和数据科学。 具体来说，一类半线性抛物方程将被考虑，重点将是确保计算的解决方案具有特别重要的属性称为时不变的最大界限原则（MBP）。这些性质对于晶粒生长和粗化、薄膜微结构、晶体生长、位错-溶质相互作用以及图像恢复和去模糊等模型都是重要的。该项目将设计和分析有效和准确的MBP保持高阶精度的时间积分方法。该项目将开发和传播软件，并为研究生提供跨学科培训的机会，拟议的活动包括计算和应用数学方面的各种研究课题，从算法设计、数值分析和有效实施到科学和工程的实际应用。具体来说，该项目提出了一个重要的一步，发展和分析高效和高阶精度的MBP保持时间积分方法。严格的分析范围内或超出目前的分析框架的一类广泛的半线性抛物方程将进行。 该项目不仅将导致数值工具和计算机代码的重大创新，用于解决这些类型的方程，但也提供了新的见解，一些突出的理论问题MBP保存和能量稳定性在时间连续和时间离散设置。 目标包括基于龙格-库塔积分因子和修正标量辅助变量法的高阶精度线性格式的设计和分析。该项目还将扩展和开发用于现有分析框架之外的一些重要相场模型的MBP保持时间积分方法，包括但不限于具有不同类型约束的质量守恒Allen-Cahn方程，对流Allen-Cahn方程和耦合Navier-Stokes/Allen-Cahn系统。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.