Savage—Hutter方程的數值格式與軟件以及三維多孔介質彈性波傳播的精確吸收邊界條件

袁禮++張文生

摘 要：泥石流/滑坡碎屑流等地質災害的沖擊力和泛濫范圍可用顆粒流薄層模型進行數值模擬研究。探索地質體內部結構可用波場探測方法。該研究給出了面向以上兩方面計算問題的階段性工作。針對顆粒流薄層近似模型Savage-Hutter方程，我們應用無振蕩中心（NOC）格式和結合MUSCL重構的LF格式，以及GPU-CUDA并行計算技術求解此方程。為了準確地模擬顆粒所受的實際摩擦力，計算中采用了靜力學條件和停止準則。還將靜力學條件和停止準則引入開源軟件titan2d中。通過數值模擬幾個簡單問題，給出了NOC格式和LF-MUSCL格式的計算精度、效率和復雜程度的比較；通過對武隆滑坡和舟曲泥石流問題的模擬，顯示了軟件titan2d計算結果的合理性，表明該軟件值得進一步發展。求解多孔介質彈性波方程在地質力學和地球物理領域中有重要應用價值，在實際計算時都必須要引入吸收邊界條件，否則會影響波場傳播的精度。該研究首次推導得到了三維多孔介質彈性波傳播的精確的吸收邊界條件。該條件由6個復雜的方程組成，而且空間是非局部的，時間是局部的，可以用標準的數值方法如有限元和有限差分法等進行計算。該結果有直接和潛在的應用價值。

關鍵詞：NOC格式 有限體積法 GPU-CUDA計算 TITAN2D軟件 多空介質彈性波 無反射邊界條件

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