Savage—Hutter方程的数值格式与软件以及三维多孔介质弹性波传播的精确吸收边界条件

袁礼++张文生

摘 要：泥石流/滑坡碎屑流等地質灾害的冲击力和泛滥范围可用颗粒流薄层模型进行数值模拟研究。探索地质体内部结构可用波场探测方法。该研究给出了面向以上两方面计算问题的阶段性工作。针对颗粒流薄层近似模型Savage-Hutter方程，我们应用无振荡中心（NOC）格式和结合MUSCL重构的LF格式，以及GPU-CUDA并行计算技术求解此方程。为了准确地模拟颗粒所受的实际摩擦力，计算中采用了静力学条件和停止准则。还将静力学条件和停止准则引入开源软件titan2d中。通过数值模拟几个简单问题，给出了NOC格式和LF-MUSCL格式的计算精度、效率和复杂程度的比较；通过对武隆滑坡和舟曲泥石流问题的模拟，显示了软件titan2d计算结果的合理性，表明该软件值得进一步发展。求解多孔介质弹性波方程在地质力学和地球物理领域中有重要应用价值，在实际计算时都必须要引入吸收边界条件，否则会影响波场传播的精度。该研究首次推导得到了三维多孔介质弹性波传播的精确的吸收边界条件。该条件由6个复杂的方程组成，而且空间是非局部的，时间是局部的，可以用标准的数值方法如有限元和有限差分法等进行计算。该结果有直接和潜在的应用价值。

关键词：NOC格式 有限体积法 GPU-CUDA计算 TITAN2D软件 多空介质弹性波 无反射边界条件

Numerical Schemes for the Savage-Hutter Equations for Granular Flows and Exact Absorbing Boundary Conditions for wave Propagation in 3D Porous Media

Yuan Li Zhang Wensheng

（Institute of Mathematics and Systems Science， Chinese Academy of Sciences）

Abstract：The impact intensities and influencing areas of debris flows in geological disasters can be simulated numerically by using thin layer models for granular flows. Interior compositions and structures below the earth surface can be explored by using wave detection methods. This report gives periodical progress towards solving the above two computational issues. For numerical solution of the Savage-Hutter equations， we apply the Non-Oscillatory Central （NOC） scheme and the LF scheme in conjunction with the MUSCL reconstruction for primitive variables， as well as GPU-CUDA parallel computing technology. Static resistance conditions and stopping criteria are implemented in order to accurately compute the friction force. They are also inserted into the open source code TITAN2D. Numerical tests against several typical examples compare the accuracy， efficiency and easiness to use between the two numerical schemes. Numerical simulations of Wulong landslide and Zhouqu debris flow disasters by using the code TITAN2D demonstrate that the numerical results are reasonable， and the code is worthy of further development. Solving the poroelastic wave equations has important application values in the area of geomechanics and geophysics. In practical computations the absorbing boundary conditions are required to increase the computational accuracy of wave propagation. We for the first time derive and obtain the exact nonreflecting boundary conditions for three dimensional poroelastic wave equations. The conditions are composed of six complicate equations and are nonlocal in space but local in time， thus they can be computed numerically with standard methods such as the finite-difference method and finite-element method. The result has direct and potential application values.

Key Words：NOC scheme； Finite volume method； GPU-CUDA computing； TITAN2D code；Poroelastic wave； Nonreflecting boundary condition