Collaborative Research: Stable and Efficient Convexity-splitting Schemes for Bistable Gradient PDEs

这项研究的目的是设计稳健、高效和实用的双稳态梯度方程的数值方法。它们形成了一类特殊的偏微分方程(PDE)，描述了材料、流体和生物研究中的重要现象。在这项工作中，PI计划详细研究斜率选择(SS)、无斜率选择(NSS)、相场晶体(PFC)和Cahn-Hilliard(CH)方程，这些方程是重要的4阶或6阶BGE，通常必须在大的空间和时间尺度上求解。这些方程(以及BGE)的数值解可能会带来巨大的挑战。在这项工作中，PI将开发BGE的凸分裂(CS)格式。CS格式在时间上是一阶或二阶精度，在空间上至少是二阶精度。它们简单、功能强大，特别适合于准确和高效地研究大型时空形态演化。一阶(时间)CS格式已经被知道了大约十年，但是，到目前为止，基本的理论还不完整，它们的应用也有一定的局限性。提出的高阶CS格式(时间上的2阶、空间上的2阶和空间上的高阶)是PI工作的新特点。所有CS格式都具有两个重要性质：无条件能量稳定和无条件唯一可解。能量稳定性经常被用来证明各种范数的稳定性，以及收敛。唯一的可解性源于格式是作为严格凸泛函的梯度导出的。因此，由于梯度下降法将无条件收敛，因此总是可以精心设计实用的求解器。这项工作的一大挑战是为潜在的高度非线性CS格式设计真正有效的求解器。在这个方向上，PI已经取得了一些早期的、重要的成功，为PFC和Cahn-Hilliard-Hele-Shaw(CHHS)方程精心制作了近乎最有效的非线性多重网格求解器。在这项工作中，他们将通过为各种BGE导出复杂、高效和时间和空间自适应的解算器来扩展这些成就。PIs将应用他们的CS格式和高效的解算器来研究薄膜粗化、肿瘤生长和治疗、两相流体流动和晶体生长等复杂的长期动力学模型。BGE允许研究人员创建大量物理和生物现象的模型，因此这项工作将对许多科学学科产生直接影响。PI将关注的特定方程(SS、NSS、PFC和CH方程)对于理解材料在原子和纳米尺度上的相变、生物生长和发育中的复杂过程以及两相流中涉及的复杂拓扑变化至关重要。举一个具体的例子，SS方程可以用来模拟某些材料表面纳米级丘陵和山谷的形成，例如用于半导体器件的那些材料。了解这些纳米结构在设备加工过程中是如何形成和移动的，对于精密制造至关重要。数学建模(例如，使用BGE)通常是进行实验室实验以找到“最佳”加工程序的更实际的替代方法。然而，在大多数实际情况下，BGE的解决方案只能使用计算机化的算法来近似。这项研究的主要目标是开发出尽可能准确、高效和健壮地逼近解的2D和3D算法。从这项工作中创建的计算机算法和源代码将适用于比本研究将探索的更一般的模型，因此将推动整个计算科学领域的发展。PI将使他们的软件包在公共领域可用，这样研究人员就可以直接访问他们的算法。除了努力实现他们的研究目标外，PI还将帮助建立和加强计算科学领域的人力资源管道，这是美国STEM教育的更广泛目标之一。研究生和本科生都将接受高性能科学计算、数值数学和建模方面的培训；他们的工作预计将成为同行评议出版物、会议演讲、技术报告和论文的基础。作为这一努力的主要组成部分，PI将通过CSUMS计划继续支持和指导达特茅斯大学的两名本科生。这些学生将接受算法和软件开发方面的实践培训。这种类型的培训在典型的本科课程中是很少见的。以这项研究为契机，PIS将致力于激励学生，特别是本科生和来自传统上代表性较低的群体的学生，在科学和工程领域追求职业生涯。

The goal of this research is to design robust, efficient, and practical numerical methods for bistable gradient equations (BGEs). These form a a special class of partial differential equations (PDEs) that describe important phenomena in materials, fluids, and biology research. For this work the PIs plan detailed investigations of the slope selection (SS), no slope selection (NSS), phase field crystal (PFC) and Cahn-Hilliard (CH) equations, which are important 4th or 6th-order BGEs that must typically be solved over large space and time scales. Numerical solution of these equations (and BGEs in general) can pose enormous challenges. In this work the PIs will develop convex splitting (CS) schemes for BGEs. CS schemes are 1st or 2nd-order accurate in time and at least 2nd-order accurate in space. They are simple, powerful, and particularly well-suited to studying large spatiotemporal morphological evolution accurately and efficiently. 1st-order (in time) CS schemes have been known for about ten years; but, up to now, the underlying theory has been incomplete and their application, somewhat limited. The proposed high-order CS schemes (2nd-order in time, 2nd-order and higher in space) are novel features of the PIs work. All CS schemes have two important properties: they are unconditionally energy stable and unconditionally uniquely solvable. The energy stability can often be exploited to prove various norm stabilities, as well as convergence. The unique solvability follows from the fact that the schemes are derived as the gradients of strictly convex functionals. As a result, practical solvers can always be crafted, since gradient descent methods will converge unconditionally. A big challenge of this work is in designing truly efficient solvers for the potentially highly nonlinear CS schemes. The PIs have had some early, important successes in this direction, having crafted nearly optimally efficient nonlinear multigrid solvers for the PFC and Cahn-Hilliard-Hele-Shaw (CHHS) equations. In this work they will extend these achievements by deriving sophisticated, efficient, and time and space adaptive solvers for a variety of BGEs. The PIs will apply their CS schemes and efficient solvers to study the complicated long-time dynamics of models for thin film coarsening, tumor growth and treatment, two-phase fluid flow, and crystal growth.BGEs allow researchers to create models of a great number of physical and biological phenomena, and hence this work will have a direct impact on many scientific disciplines. The specific equations that the PIs will focus on (SS, NSS, PFC, and CH equations) are vital for understanding phase transformations of materials at the atomic and nanometer scales, the complex processes in biological growth and development, and the complicated topological change involved in two-phase flows. For a specific example, the SS equation can be used model the formation of nano-scopic hills and valleys on the surfaces of certain materials, such as those used in semiconductor devices. Knowing how these nano-structures form and move during device processing is critical for precise manufacture. Mathematical modeling (using BGEs, for example) is often a more practical alternative to doing laboratory experiments to find ``optimal" processing procedures. However, in most practical situations, solutions to BGEs can only be approximated using computerized algorithms. The primary goal of this research is to develop 2D and 3D algorithms that approximate the solutions as accurately, efficiently, and robustly as possible. The computer algorithms and source codes created from this work will apply to even more general models than will be explored in this research and will therefore advance the field of computational science as a whole. The PIs will make their software packages available in the public domain so that researchers will have direct access to their algorithms. In addition to working toward their research goals, the PIs will help to build and reinforce the human resources pipeline in the field of computational sciences, which is one of the broader goals in STEM education in the US. Both graduate and undergraduate students will receive training in high-performance scientific computing, numerical mathematics, and modeling; and their work is expected to form the bases of peer-reviewed publications, conference talks, technical reports, and theses. As a major component of this effort, the PIs will continuously support and mentor two UMass, Dartmouth undergrads through the CSUMS program. These students will get hands-on training in algorithm and software development. This type of training is rare in the typical undergraduate curriculum. Using this research as a venue, the PIs will work to inspire students, especially undergraduates and those from traditionally underrepresented groups, to pursue careers in science and engineering.