泥石流动问题的非结构网格有限体积算法研究

Debris flow is a type of greatly destructive geological hazard, and our country is one of the debris flow disaster prone areas. For many potential debris flows, it is almost impossible to completely rely on engineering measures for prevention and treatment. However, numerical simulation for the dynamic process of debris flow can not only inverse and reproduce the full process of occurrence and development for this disaster, thus improving people's understanding of debris flow disaster law, but also can carry out quantitative risk assessment to provide scientific basis for prevention, planning and control of debris flow disaster.This project will study systematically the debris flow dynamics model which is based on the continuum, explore the finite volume method on unstructured mesh for simulating the debris flow over complex terrain, and develop the related high-efficient simulation technologies such as the adaptive algorithm. Then the algorithm analysis theory of debris flow disaster is established. Further, 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 problem that it is difficult to simulate the whole process of debris flow in current commercial software and related literature is solved, which provides a reliable technical means for debris flow disaster prediction and prevention and also provides important theoretical support for its risk assessment and prevention of this disaster.

泥石流是一种破坏性极大的地质灾害类型，我国是泥石流灾害多发地区之一。对许多潜在的泥石流灾害，完全依赖于工程防治几乎是不肯能的。然而，基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全过程，从而提高人们对泥石流灾害规律的认识，还能对其进行定量风险评估，为防治、规划及治理泥石流灾害提供科学依据。本项目将系统研究基于连续介质的泥石流动力学模型，探索适合复杂地形地貌的泥石流模拟的非结构网格的有限体积法，发展其自适应算法等相关高效模拟技术，并建立泥石流灾害的算法分析理论。通过算法的程序化再现，将模拟出泥石流灾害的动力学过程，预测潜在的灾害范围。通过本项目的研究，解决目前商业软件和相关文献中泥石流全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，也为其风险评估和防治提供重要理论支撑。

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。本项目是在北京大学李若教授的精心指导下开展的，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。