Numerical Improvements, Mesh Adaptation and Parameter Identification for Parallel Finite Element Stokes Ice Sheet Modeling

由于陆冰在全球海平面和气候系统的其他部分中起着至关重要的作用，对陆冰演化的数值模拟一直是人们越来越感兴趣的主题。非线性三维Stokes流是冰盖动力学概念模型中的黄金标准。目前广泛使用的浅冰、浅陆架、L1L2和高阶近似都是通过尺度分析作为3D Stokes模型的简化形式获得的，但在许多情况下，伴随着保真度的损失。PI与一组合作者密切合作，初步开发了用于冰盖模拟的并行有限元非线性3DStokes动力核心。该项目的目标是通过进一步研究和提高现有的有限元Stokes冰盖模型的效率、精度、可用性和稳健性来改进该模型。PI将首先研究与有限元Stokes冰盖动力学求解器有关的一些问题，包括分析和实现基于牛顿的快速迭代方法来处理流变学和基本边界条件的非线性，以及用于增强数值模式中的局部守恒性的自适应混合离散化方案。在我们考虑的冰盖模式中，Stokes冰盖动力学方程完全耦合到温度演化方程，因此需要稳定和准确的有限元温度求解器，其精度与Stokes求解器相当，也将被开发。众所周知，利用伴随方程方法可以直接得到感兴趣量的精确解。PI还将研究和开发基于伴随方程的方法，使用面向目标的优化方法进行自适应网格加密和识别基本边界滑动参数。尽管近年来数值冰盖模式得到了稳步改进，但要使其在长时间和整个冰盖尺度上更加可靠、高效和可用，仍需进行大量工作。改进的Stokes冰盖数值模式将通过使用高精度的自适应有限元离散格式、高度可扩展的并行线性和非线性系统求解器、面向目标的变分辨率网格划分策略以及模型参数的有效逆设计来实现高效率和高精度。这项拟议的调查将通过数值模拟为理解陆地冰的演化提供新的见解。国际和平研究所将不仅向该领域的研究人员，而且也向对数值方法和计算地球物理感兴趣的更广泛的社区，通过出版物、出席会议、维护信息丰富的网站，积极传播他的研究成果和经过测试的软件。该项目的潜在影响是非常巨大的。拟议项目产生的直接和变革性创新将极大地提高气候系统模拟中的计算冰盖模型能力。此外，该项目还将为对计算和应用数学感兴趣的研究生提供一个独特的教育机会，让他们参与一个结合了数学、计算机科学和地质科学的跨学科研究计划。

The numerical modeling of land ice evolution has been a subject of growing interest because of the crucial role land ice plays in global sea level and other parts of the climate system. Nonlinear 3D Stokes flow is the gold standard among conceptual models for ice sheet dynamics. The current widely-used shallow-ice, shallow-shelf, L1L2, and higher-order approximations are all obtained as reduced forms of the 3D Stokes model by means of scaling analysis, but in many situations, with an attendant loss of fidelity. The PI has closely collaborated with a team of collaborators on the preliminary development of a parallel finite element nonlinear 3D Stokes dynamical core for ice sheet modeling. The goal of the proposed project is to advance the current finite element Stokes ice sheet model by further studying and enhancing its efficiency, accuracy, usability, and robustness. The PI will first investigate some issues related to the finite element Stokes ice sheet dynamics solver, including analysis and implementation of Newton-based fast iterative methods for treating both rheological and basal boundary condition nonlinearities and an adaptive hybrid discretization scheme for enhancing the local conservation properties in our numerical model. In the ice-sheet model we consider, the Stokes ice sheet dynamics equations are fully coupled to the equation for temperature evolution, thus stable and accurate finite element temperature solver with accuracy commensurate with that of the Stokes solver is desired and will also be developed. It is well-known that the adjoint equation approach allows one to directly obtain accurate solutions for the quantity of interests. The PI will also investigate and develop adjoint equation-based methods for adaptive mesh refinement and identification of basal boundary sliding parameter using goal-oriented optimization approaches.Although numerical ice sheet models have steadily improved in recent years, much work is needed to make them more reliable, efficient and usable at long time and whole ice sheet scales. The enhanced numerical Stokes ice sheet model will achieve high degrees of efficiency and accuracy through the use of high-order accurate adaptive finite element discretization schemes, highly scalable parallel linear and nonlinear system solvers, goal-oriented variable resolution meshing strategies, and effective inverse design for model parameters. The proposed investigation would offer new insights through numerical simulations to the understanding of land ice evolution. The PI will actively disseminate his research results and tested software not only toresearchers in the area but also to much broader communities with interests in numerical methods and computational geophysics through publications, attending meetings, maintaining an informative web-site. The potential impact of the project is very substantial. Direct and transformative innovations resulting from the proposed project will greatly improve computational ice sheet model capabilities in the climate system modeling. In addition, this project will also offer a unique educational opportunity for graduate students with interests in computational and applied mathematics by having them participate in an interdisciplinary research program that combines mathematics, computer science and geological sciences.