RII Track-4:NSF: Construction of New Additive and Semi-Implicit General Linear Methods

复杂动力学现象的建模在科学和工程的许多领域都是至关重要的。特别是，数值模拟为我们提供了更好的理解自然过程，并在工程系统的设计和优化中发挥着至关重要的作用。在许多情况下，这些模拟需要时间相关的偏微分方程的解决方案。由于存在广泛的空间和时间尺度，求解这样的方程是具有挑战性的。因此，有一个持续的需要，精心设计的时间积分算法，使建模人员能够更有效，更准确地模拟复杂的物理系统。这个EPSCoR RII轨道-4奖提供了奖学金，以助理教授杜兰大学和支持研究生。它将加强杜兰大学和洛斯阿拉莫斯国家实验室之间的合作，使多次访问与领先的数学家和科学家合作和互动。该项目将开发新的隐式一般线性方法，用于解决偏微分方程离散化产生的刚性系统。这些方法将被设计为添加和非线性分区的问题，允许隐式分布在最计算有效的方式。此外，该项目将研究不同类型的内隐，包括可以利用预处理现代发展的全内隐方法。方法设计将借鉴最近推出的时间积分框架，利用插值多项式，大大简化了多值积分器的建设的想法。最后，作为这项工作的一部分开发的所有方法将被验证和应用，以解决等离子体，辐射传输和流体动力学领域出现的一系列实际问题。该奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

The modeling of complex dynamical phenomena is critical to many fields of science and engineering. In particular, numerical simulations provide us with a greater understanding of natural processes and play an essential role in the design and optimization of engineering systems. In many cases, these simulations require the solution of time-dependent partial differential equations. Solving such equations is challenging due to the presence of wide-ranging spatial and temporal scales. There is therefore a continuing need for carefully designed time integration algorithms that allow modelers to simulate complex physical systems more efficiently and accurately.This EPSCoR RII Track-4 award provides a fellowship to an Assistant Professor at Tulane University and support for a graduate student. It will strengthen a collaboration between Tulane University and Los Alamos National Laboratory by enabling multiple visits to collaborate and interact with leading mathematicians and scientists. The project will develop new implicit general linear methods for solving stiff systems arising from the discretization of partial differential equations. These methods will be designed for both additively and nonlinearly partitioned problems, allowing the implicitness to be distributed in the most computationally efficient way. Moreover, the project will investigate differing types of implicitness, including fully-implicit methods that can leverage modern developments in preconditioning. Method design will draw on ideas from a recently introduced time integration framework that utilizes interpolating polynomials to greatly simplify the construction of multivalued integrators. Lastly, all the methods developed as part of this work will be validated and applied to solve a range of practical problems arising from the fields of plasmas, radiative transport, and fluid dynamics.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.