SI2-SSE: Development and Implementation of Software Elements using State-of-the-Art Computational Methodology to Advance Modeling Heterogeneities and Mixing in Earth's Mantle

该项目涉及科学软件元素（SSE）的开发和实施，基于现代高分辨率数值方法，用于在存在热对流的情况下对粘性流体中材料特性的陡峭梯度和尖锐界面进行建模。该项目的目标是解决地球动力学的迫切需要，其中连续介质力学应用于地球物理过程的研究，如地球对流？的斗篷。地球动力学研究的一个主要工具是数亿至数十亿年地球内部极粘流体流动的计算模型。这些模型面临的一个长期挑战是需要准确地模拟温度，粘度和其他属性的尖锐界面。例如，当模拟俯冲（冷的构造板块陷入热的内部）或上升的羽流（热的边界层不稳定性通过地幔上升并遇到构造板块的冷边界层）时，就会出现这些问题。该项目将促进跨学科交流，并将最先进的应用数学和计算数学应用于物理学的基本问题。它涉及早期职业数学科学家在应用国家的最先进的数值算法的地球动力学，特别是，将提供一个机会，增加妇女参与数学和地球动力学研究。该项目涉及设计和实施国家的-用于计算地球地幔中重要过程的演化的技术SSE，其中问题的基本特征是存在一个或多个移动的边界、界面或温度、成分或粘度的陡峭梯度。SSE将解决目前限制现代地幔对流模拟的两个关键问题。目前使用的所有地幔对流的计算模型在温度和粘度的急剧梯度附近产生显著的过冲和下冲。这些过冲和下冲的原因是数值伪影，这在其他领域（如计算冲击物理界）是众所周知的。在过去的三十年里，计算激波物理的研究人员已经开发了各种高阶精确的单调数值方法，这些方法保留了计算量的物理正确的最大值和最小值，同时产生这些量的高阶精确数值近似。计算地球动力学的另一个迫切需要是跟踪物质成分等数量的不连续跳跃的能力。这里需要高阶精确的界面跟踪算法，因为这些领域经历大规模的变形，但诸如粘度的量必须在两种材料之间的界面处精确地近似。

This project involves the development and implementation of scientific software elements (SSEs), based on modern, high-resolution numerical methods for modeling steep gradients and sharp interfaces of material properties in viscous fluids in the presence of thermal convection. The goal of this project is to address a compelling need in geodynamics, in which continuum mechanics is applied to the study of geophysical processes, such as convection in the Earth?s mantle. A primary tool of geodynamics research is computational models of the flow of the extremely viscous interior of the Earth over hundreds of millions to billions of years. A long-standing challenge for these models is the need to accurately model sharp interfaces in temperature, viscosity, and other properties. These arise when, for example, modeling subduction (in which a cold tectonic plate plunges into the hot interior) or rising plumes (in which a hot boundary layer instability rises through the mantle and encounters the cold boundary layer of the tectonic plates). The project will foster interdisciplinary communication and the application of state-of-the-art applied and computational mathematics to fundamental problems in geophysics. It involves early-career mathematical scientists in the application of state-of-the-art numerical algorithms to geodynamics and, in particular, will provide an opportunity to increase the participation of women in mathematics and geodynamics research.This project involves the design and implementation of state-of-the-art SSEs for computing the evolution of significant processes in the Earth's mantle in which an essential feature of the problem is the presence of one or more moving boundaries, interfaces, or steep gradients in temperature, composition, or viscosity. The SSEs will address two critical issues that currently limit modern mantle convection simulations. All computational models of mantle convection currently in use produce significant overshoot and undershoot in the neighborhood of sharp gradients in temperature and viscosity. The cause of these overshoots and undershoots is a numerical artifact, which is well-known and well-understood in other fields, such as the computational shock physics community. Over the past thirty years researchers in computational shock physics have developed a variety of high-order accurate, monotone numerical methods, which preserve the physically correct maximum and minimum values of the computed quantities, while producing a high-order accurate numerical approximation of these quantities. Another compelling need in computational geodynamics is the ability to track discontinuous jumps in quantities such as material composition. Here high-order accurate interface tracking algorithms are required, since these fields undergo large-scale deformation, yet quantities such as the viscosity must be accurately approximated at the interface between two materials.