VTI介质qP波方程高精度有限差分算子

波动方程有限差分法是一种使用广泛的地震波数值模拟方法.但是有限差分法本身固有存在着数值频散问题,会降低地震波场模拟的精度与分辨率.为了克服常规有限差分算子的数值频散,本文针对VTI介质地震波数值模拟问题,构造了频率-空间域qP波波动方程高精度有限差分优化算子,根据最优化理论中高斯-牛顿法确定了高精度有限差分算子的优化系数.利用常规差分算子和高精度优化差分算子对归一化相速度的频散关系精度进行了对比分析,并对均匀各向同性介质和均匀VTI介质中的qP波地震波场进行了有限差分数值模拟,通过频散关系精度分析和波场数值模拟结果表明：有限差分优化算子具有较高的波场数值模拟精度,有效压制了传统有限差分算子数值模拟中的数值频散现象,提高了有限差分算子精度,为VTI介质频率-空间域qP波正演模拟奠定了基础.     

Error analysis of finite difference methods for two-dimensional advection–dispersion–reaction equation

In this paper, the numerical errors associated with the finite difference solutions of two-dimensional advection–dispersion equation with linear sorption are obtained from a Taylor analysis and are removed from numerical solution. The error expressions are based on a general form of the corresponding difference equation. The variation of these numerical truncation errors is presented as a function of Peclet and Courant numbers in X and Y direction, a Sink/Source dimensionless number and new form of Peclet and Courant numbers in X–Y plane. It is shown that the Crank–Nicolson method is the most accurate scheme based on the truncation error analysis. The effects of these truncation errors on the numerical solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution for predicting contaminant plume distribution in uniform flow field. Considering computational efficiency, an alternating direction implicit method is used for the numerical solution of governing equation. The results show that removing these errors improves numerical result and reduces differences between numerical and analytical solution.     

A central difference method with low numerical dispersion for solving the scalar wave equation

In this paper, we propose a nearly‐analytic central difference method, which is an improved version of the central difference method. The new method is fourth‐order accurate with respect to both space and time but uses only three grid points in spatial directions. The stability criteria and numerical dispersion for the new scheme are analysed in detail. We also apply the nearly‐analytic central difference method to 1D and 2D cases to compute synthetic seismograms. For comparison, the fourth‐order Lax‐Wendroff correction scheme and the fourth‐order staggered‐grid finite‐difference method are used to model acoustic wavefields. Numerical results indicate that the nearly‐analytic central difference method can be used to solve large‐scale problems because it effectively suppresses numerical dispersion caused by discretizing the scalar wave equation when too coarse grids are used. Meanwhile, numerical results show that the minimum sampling rate of the nearly‐analytic central difference method is about 2.5 points per minimal wavelength for eliminating numerical dispersion, resulting that the nearly‐analytic central difference method can save greatly both computational costs and storage space as contrasted to other high‐order finite‐difference methods such as the fourth‐order Lax‐Wendroff correction scheme and the fourth‐order staggered‐grid finite‐difference method.     

Optimal weighting in the finite difference solution of the convection-dispersion equation

Oscillation and numerical dispersion limit the reliability of numerical solutions of the convection-dispersion equation when finite difference methods are used. To eliminate oscillation and reduce the numerical dispersion, an optimal upstream weighting with finite differences is proposed. The optimal values of upstream weighting coefficients numerically obtained are a function of the mesh Peclet number used. The accuracy of the proposed numerical method is tested against two classical problems for which analytical solutions exist. The comparison of the numerical results obtained with different numerical schemes and those obtained by the analytical solutions demonstrates the possibility of a real gain in precision using the proposed optimal weighting method. This gain in precision is verified by interpreting a tracer experiment performed in a laboratory column.     

Estimating finite difference block equivalent hydraulic conductivity for numerically solving the Richards' equation

A method is proposed for calculating the equivalent hydraulic conductivity (EHC) within a finite difference block (FDB). Application of the constant‐flux assumption of Darcy's Law, the EHC equals to the integration of effective hydraulic conductivity (K_w) as a function of pressure head (h_w) divided by the head difference at the ends of the FDB. Error analysis show that the constant‐flux (CF) EHC estimates are better than those computed by the commonly used arithmetic‐mean (AM), geometric‐mean (GM), and harmonic‐mean (HM) techniques. CF EHC results are even more superior at larger interblock head difference situations. Simulations of water infiltration experiments show that simulations using the CF EHC or AM or GM weighting technique have only slight difference while applying the Neumann type boundary condition at the ground surface. In case of the Dirichlet type boundary condition, however, the CF EHC is superior to the other two in correctly estimating the depth of infiltration while enlarging the grid size. Therefore, it is recommended to adopt the CF EHC with a larger grid size to the more stable and more efficient results. Copyright © 2007 John Wiley & Sons, Ltd.     

有限元算法在声波方程数值模拟中的频散分析

针对有限元算法在地震波数值模拟中的数值频散问题,利用集中质量矩阵双线性插值有限元算法,推导了二维声波方程的频散函数.在此基础上采用定量分析方法,对比分析了网格纵横长度比变化时的入射方向、空间采样间隔、地震波频率以及地层速度对数值频散的影响.数值算例和模型正演结果表明：当采用集中质量矩阵双线性插值有限元算法时,为了有效地压制数值频散,在所使用震源子波的峰值频率对应的波长内,采样点数目应不少于20个;减小网格长度的纵横比可以有效地抑制入射角(波传播方向与z轴的夹角)较小的地震波的数值频散;地震波频率越高,传播速度越慢,频散越严重,尤其是当相速度与其所对应的频率比值小于2倍空间采样间隔时,不仅会出现严重的数值频散,还会出现假频现象.     

A finite volume upwind scheme for the solution of the linear advection–diffusion equation with sharp gradients in multiple dimensions

A finite volume upwind numerical scheme for the solution of the linear advection equation in multiple dimensions on Cartesian grids is presented. The small-stencil, Modified Discontinuous Profile Method (MDPM) uses a sub-cell piecewise constant reconstruction and additional information at the cell interfaces, rather than a spatial extension of the stencil as in usual methods. This paper presents the MDPM profile reconstruction method in one dimension and its generalization and algorithm to two- and three-dimensional problems. The method is extended to the advection–diffusion equation in multiple dimensions. The MDPM is tested against the MUSCL scheme on two- and three-dimensional test cases. It is shown to give high-quality results for sharp gradients problems, although some scattering appears. For smooth gradients, extreme values are best preserved with the MDPM than with the MUSCL scheme, while the MDPM does not maintain the smoothness of the original shape as well as the MUSCL scheme. However the MDPM is proved to be more efficient on coarse grids in terms of error and CPU time, while on fine grids the MUSCL scheme provides a better accuracy at a lower CPU.     

横向各向同性介质紧致交错网格有限差分波场模拟(英文)

针对有限差分数值模拟的频散问题,本文将交错网格技术和紧致差分格式相结合,推导了横向各向同性介质一阶速度一应力波动方程的紧致交错网格差分格式;对比分析了紧致交错网格差分格式、交错网格差分格式以及紧致差分格式的截断误差主项,并利用Fourier误差分析方法分析了上述三种差分格式的近似精度;在此基础上,分别采用上述三种差分格式进行了波场数值模拟。结果表明,当差分方程阶数相同时,紧致交错网格差分格式截断误差最小,数值频散最弱,差分精度最高,证实了该方法的有效性。     

Evaluation of the Ross fast solution of Richards’ equation in unfavourable conditions for standard finite element methods

Ross [Ross PJ. Modeling soil water and solute transport – fast, simplified numerical solutions. Agron J 2003;95:1352–61] developed a fast, simplified method for solving Richards’ equation. This non-iterative 1D approach, using Brooks and Corey [Brooks RH, Corey AT. Hydraulic properties of porous media. Hydrol. papers, Colorado St. Univ., Fort Collins; 1964] hydraulic functions, allows a significant reduction in computing time while maintaining the accuracy of the results. The first aim of this work is to confirm these results in a more extensive set of problems, including those that would lead to serious numerical difficulties for the standard numerical method. The second aim is to validate a generalisation of the Ross method to other mathematical representations of hydraulic functions.     

On numerical errors associated with the iterative alternating direction implicit (Iadi) finite difference solution of the two dimensional transient saturated-unsaturated flow (Richards) equation

Numerical models for simulation of multidimensional unsaturated flow are becoming increasingly available, but relatively little has been reported on the detailed analysis of numerical errors associated with such schemes. For unsaturated-saturated flow, further complexity is introduced to the highly non-linear unsaturated problem as the form of governing equation changes within the flow field. In this paper, two-dimensional simulation of infiltration to a water table is considered. Numerical errors associated with selection of time step, grid geometry, convergence criteria, and the representation of internodal hydraulic conductivity are discussed with respect to moisture content profiles and mass balance error. Although solution sensitivity to numerical parameters is problem specific, the results presented indicate the nature and magnitude of numerical effects which should not be overlooked in model applications.     

The impact of boundary on the fractional advection–dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives

The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl⁻ in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.