泥石流动问题的非结构网格有限体积算法研究

Debris flow is a kind of geological disaster phenomenon that often occurs in mountainous areas and is extremely destructive. It is impossible to take engineering measures to prevent and control debris flow completely. However, the numerical simulation of the dynamic process of debris flow can not only inverse and reproduce the occurrence and development process of debris flow disaster, but also raise our awareness for the development law. Therefore, it is of great significance to carry out the research of the numerical simulation for the dynamic process of debris flow, which will provide scientific reference for organizing disaster prevention and mitigation. The project will systematically study the debris flow dynamics model based on the continuum, then explore the finite volume method and adaptive algorithms under unstructured mesh, develop numerical methods suitable for rapid simulation of the debris flow over complex terrain, and establish corresponding algorithm analysis theories. With the programming reproduction of the algorithms, the full dynamic process for debris flow disaster can be simulated out, which is helpful to estimate the potential scope of disaster. Through the research of this project, the difficult problem about that currently it is hard to simulate the dynamic process of debris flow can be solved, providing not only a reliable technical means for predicting, preventing and controlling the debris flow disaster, but also an important theoretical support for risk assessment of this disaster. Meanwhile, through the implementation of the project, conditions will be provided for the cultivation of young people in the mathematics discipline of the Three Gorges University. Thus, it promotes the development of the mathematics discipline of the Three Gorges University and changes the current status of unbalanced development in the mathematics discipline in China to a certain extent.

泥石流是一种常发生于山区，破坏性极大的地质灾害现象，完全采用工程措施来防治几乎不可能。由于泥石流动力过程的数值模拟不但可以反演、再现泥石流灾害发生和发展的复杂过程，且有助于提高人们对泥石流灾害发展规律的认识，因此开展泥石流动力过程的数值模拟研究，对组织防灾减灾提供科学参考具有十分重要的意义。本项目将系统研究基于连续介质的泥石流动力学模型，设计非结构网格下的有限体积算法和自适应算法，发展适合快速模拟复杂地形地貌下泥石流动的数值方法，并建立相应的算法分析理论。通过算法的程序化再现，模拟泥石流灾害的动力过程，预测可能受到灾害的泛滥范围。本项目的研究成果将解决泥石流全过程难以模拟的问题，为潜在大型泥石流灾害提供预测手段，为泥石流全过程的风险评估和防治提供基础理论支撑。同时，本项目还将为三峡大学数学学科培养青年学者提供条件，促进三峡大学数学学科的发展，在一定程度改变我国数学学科地区发展不平衡的现状。

泥石流、滑坡等是破坏性极大的地质灾害类型，目前对这些灾害的预测和防治十分必要。本项目主要通过数值模拟的方法来认识泥石流、滑坡的运动规律。项目双方紧密合作，首先研究了Savage-Hutter模型的一些数学性质，发现该模型不满足全局双曲性和旋转不变性，这为在非结构网格上求解该模型带来了很大的困难，但是，实际的泥石流、滑坡等多发生在地形复杂的山区，在非结构网格上求解优势突出。为此，在非结构三角形网格上，设计了一种二阶精度的有限体积格式，使其能有效地求解该模型，且数值结果与文献结果相吻合。其次，基于WENO重构思想，进一步研究了基于结构网格的高精度数值格式，并对其在三种不同数值通量下的数值效果进行了比较。最后，为了研究高效高精度数值算法，基于精细积分、高阶紧致差分格式、分裂算法和Hopf-Cole变换，构造了一种能高精度求解多维Burgers方程组的数值算法。部分结果将发表在国际SCI期刊《Numerical Mathematics: Theory, Methods and Applications》和《Applied Mathematics and Computation》上。下一步将在间断Galekrin方法的框架下，重点研究具有与未知函数导数相关的间断系数问题的求解。本项目的研究成果有效地解决了泥石流动力过程难以模拟的问题，为泥石流灾害预测和防治提供一种可靠的技术手段。同时，通过近一年的访问学习，进一步提升了来访者三峡大学张小华博士的学术水平，开拓了他的研究视野，丰富了他的研究经验，也必将促进三峡地区数学学科的进一步发展。