High-order temporal and implicit spatial staggered-grid finite-difference operators for modelling seismic wave propagation
High-order temporal and implicit spatial staggered-grid finite-difference operators for modelling seismic wave propagation
复制标题
用于模拟地震波传播的高阶时间和隐式空间交错网格有限差分算子
DOI：
10.1093/gji/ggz059
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发表时间：
2019-05-01
影响因子：
2.8
通讯作者：
Li, Zhenchun
中科院分区：
文献类型：
--
作者：
Ren, Zhiming;Li, Zhenchun
有限差分方法被广泛应用于声波和弹性波方程的数值求解。与时间二阶差分方法相比，时间高阶差分方法具有更好的精度和稳定性。此外，空间导数的隐式计算也可以带来显著的精度提高。现有的具有高精度时间精度的隐式FD方法都是基于中心网格的。在本文中，我们提出了一种隐式交错网格FD(SFD)格式，其中组合模板是菱形/棱锥体和交叉模板在二维/三维情况下的组合，用于模拟标量波的传播。该方案基于组合模板和传统模板，分别使用高阶时间和隐式空间FD算子计算时间和空间导数。我们推导了二维和三维标量波动方程的FD格式的色散关系，并用泰勒级数展开(TE)和最小二乘法(LS)估计了时间和隐式空间FD系数。根据FD系数的类型，我们构造了四个隐式SFD算子：TE-TE、TE-LS、LS-TE和LS-LS算子。我们与现有的几种SFD格式进行了比较：传统的显式和隐式格式、最优显式格式和隐式和显式时间高阶格式。二维和三维色散分析、稳定性分析和模拟算例表明，我们的隐式格式比其他格式具有更高的精度，对稳定性条件的要求比显式时间高阶SFD格式略严格。由于具有更高的精度，我们的带有LS-LS算子的隐式SFD格式允许更短的FD算子和更大的网格间距，从而可以提高计算效率。
Finite-difference (FD) methods are widely used for numerical solution of acoustic and elastic wave equations. Temporal high-order FD methods exhibit better accuracy and stability than the methods with second-order differencing in time. Also, the implicit calculation of spatial derivatives can bring significant improvement in accuracy. The present implicit FD methods with high-order accuracy in time are based on centred grids. In this paper, we propose an implicit staggered-grid FD (SFD) scheme with a combined stencil, which is the combination of rhombus/pyramid and cross stencils in 2-D/3-D case, for modelling scalar wave propagation. Our scheme computes the temporal and spatial derivatives using high-order temporal and implicit spatial FD operators based on the combined stencil and the conventional stencil, respectively. We derive the dispersion relations of the FD scheme for 2-D and 3-D scalar wave equations and estimate temporal and implicit spatial FD coefficients by Taylor series expansion (TE) and least squares (LS). According to the kinds of FD coefficients, we formulate four implicit SFD operators: TE-TE, TE-LS, LS-TE and LS-LS operators. We carry out the comparison between our scheme and several existing SFD schemes: the conventional explicit and implicit, optimal explicit and implicit and explicit temporal high-order schemes. 2-D and 3-D dispersion analysis, stability analysis and modelling examples reveal that our implicit scheme has greater accuracy than other schemes and requires slightly stricter stability condition than the explicit temporal high-order SFD schemes. Owing to higher accuracy, our implicit SFD scheme with LS-LS operators allows for shorter FD operators and larger grid spacing, which can increase the computational efficiency.