Efficient dynamic mesh adaptation for numerical simulation of evolutionary problems arising from physical science

本研究项目旨在开发高效的动态网格自适应策略，用于物理科学演化问题的数值模拟。网格自适应是科学和工程中基于网格的数值模拟中不可缺少的工具。其基本思想是将更多的网格节点放置在解变化较大的区域，而不是解平滑的区域。通过这种方式，需要更少的网格节点来达到指定的精度水平，从而获得显著的经济效果。动态网格自适应是网格自适应的一种，它通过移动网格节点来跟踪物理问题的动态特征。其在时间上的连续性使得动态网格自适应方法成为进化问题数值求解的自然选择。研究内容包括运动网格上偏微分方程的时间积分的研究，Schwarz波形移动网格方法的发展，以及网格方程简单而有效的求解器的发展。所有的研究都将集中在由液晶模型和相变问题产生的演化偏微分方程组。这些问题有许多重要的工业应用，引起了科学家们的极大兴趣。该项目的成功完成将为研究由这些问题引起的解的奇异性的形成和运动界面的传播提供一个强有力的工具。研究生将积极参与这项研究项目。学生的训练将受益于大规模的计算和理论研究。

This research project is to develop efficient dynamic mesh-adaptation strategies for the numerical simulation of evolutionary problems arising from physical science. Mesh adaptation is an indispensable tool for use in mesh-based numerical simulation in science and engineering. Its basic idea is to put more mesh nodes in regions of large solution variation than those where the solution is smooth. In this way, fewer mesh nodes are required to reach a specified level of accuracy and thus significant economies are gained. Dynamic mesh adaptation is a type of mesh adaptation which moves mesh nodes around to follow the dynamic features of the physical problem. Its continuous nature in time makes a dynamic mesh-adaptation method the natural choice of method for use in the numerical solution of evolutionary problems.The proposed research focuses on improving the efficiency of dynamic mesh adaptation. Topics of study include the investigation of time integration of partial differential equations on moving meshes, the development of the Schwarz waveform moving-mesh method, and the development of simple and efficient solvers for mesh equations. All studies will be targeted on evolutionary partial differential equations arising from liquid-crystal models and phase-change problems. These problems have many important industrial applications and have attracted considerable interest among scientists. Successful completion of this project will provide a powerful tool for studying the formation of solution singularities and the propagation of moving interfaces arising from these problems. Graduate students will be actively involved in the research project. Students' training will benefit from large-scale computations as well as theoretical studies.