全波场地震波数值模拟中的边界处理实践研究

在用有限差分法或有限元法模拟无界区域中的波动时,需要对计算区域的边界做特殊处理,以消除由于把地震波的传播设定在有限区域而产生的边界反射。为了这一目的,人们研究出了多种人工边界处理方法,完全匹配层（PML）吸收边界条件就是理想的方法之一,现已被广泛应用。本文将PML吸收边界条件应用于全波场地震波的数值模拟,数值计算实验表明,对qP波,匹配层的厚度为5个网格间距即可达到要求,而对qSV波与qSH波,为达到理想的吸收效果,匹配层的厚度应当增大,当厚度为13个网格间距时达到了理想的吸收效果。     

一种实用的水波数值模拟方法

建立了一种实用的水波动态数值模拟方法,它是基于达朗贝尔粘弹性介质的理论,构建了二阶中心网格有限差分水波正演递推算子,并将常规波场的频散现象作为水波的涌浪特征而得到有效利用,同时构建了多种真实、动感且实用的涌浪波场传播场景。数值计算表明,通过调节该水波离散系统的模拟参数可以有效模拟多种复杂环境下的水波传播,从而更好地为水利建设服务。     

基于GPU的波动方程正演模拟的实现

随着计算机技术的发展,使得波动方程正演由理论研究应用到实际地震勘探中成为了可能。而有限差分技术作为地震波场模拟的一种有效数值方法,它具有实现简单,速度快,从而被广泛应用正演计算密集的波形正反演中。地震波正演的计算量大,通过CPU来计算地震波正演模拟严重影响整体运算效率,GPU通用计算技术的产生及其在内的数据并行性有望改变这一状况。该文主要研究波动方程正演在GPU上的模拟实现。     

一类Lagrange坐标系下的ENO有限体积格式

本文首先从积分形式的二维Lagrange流体力学方程组出发,使用ENO高阶插值多项式,推广了四边形结构网格下的一阶有限体积格式,构造得到了一类结构网格下的高精度有限体积格式.该格式针对单介质问题具有良好的计算效果,同时在处理多介质问题时,不会产生物质界面附近强烈的震荡.结合有效的守恒重映方法,用ALE方法进行数值模拟,得到了预期的效果.     

两类相场方程的紧致指数时间差分法设计

本文针对相场方程提出稳定的高阶紧致指数时间差分算法.该算法具有完全显式的特性,从而避免了求解线性或非线性方程组.算法使用精确指数时间差分和多步法近似以保证精确性;通过线性算子分裂控制刚性非线性项以增强稳定性;同时引入有限差分格式的紧致表示大大降低了指数时间差分法的存储需求和计算量.算法的精确性和高效性通过CahnHilliard方程和Willmore问题相场模型的大规模三维模拟进行了验证.     

基于FDTD的平面波入射介质的仿真

在二维时域有限差分法的基础上,以MATLAB为环境,进行电磁场数值仿真计算以及设置入射波与介质圆柱边界条件.用完美匹配层、矢量亥姆霍兹方程解决了吸收边界和平面波入射有损介质的问题.将平面电磁波入射到介质圆柱上的状态很好地模拟出来.     

一阶、二阶导数耦合的紧致差分格式及其应用

在数值计算中可能遇到求解一阶导数和二阶导数耦合的微分方程,为了能用紧致差分格式进行计算,针对这样的方程,建立了考虑一阶、二阶导数耦合的紧致差分格式,利用这一方法可以直接对方程进行离散求解。通过具体算例,验证该类紧致差分格式的优越性,还将这类紧致差分格式运用到求解二维偏微分方程中。     

二维波动方程的一种高精度紧致差分方法

提出了一种求解二维波动方程的高精度紧致差分方法,该方法首先利用紧交替方向隐式差分格式,其截断误差为O(τ2+h4),分别在粗网格和细网格上对原方程进行求解,然后利用Richardson外推计算一次,进一步提高精度,得到了二维波动方程具有O(τ4+h6)精度的数值解。数值实验验证了该方法的可靠性、有效性和精确性。     

一维非定常对流扩散方程的高阶组合紧致迎风格式

通过将对流项采用四五阶组合迎风紧致格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的半离散格式在时间方向采用四阶龙格库塔方法求解,从而得到了一种求解非定常对流扩散方程问题的高精度组合紧致有限差分格式,其收敛阶为O(h~4+τ~4).经Fourier精度分析和数值验证,证实了格式的良好性能.三个数值算例包括线性常系数问题,矩形波问题和非线性问题,数值结果表明:该格式具有很高的分辨率,且适用于对高雷诺数问题的数值模拟.     

基于物理的烟雾动画

基于物理的烟雾模拟中,需要求解Navier-Stokes偏微分方程组．通过量纲分析,简化了该方程．在求解对流项时采用半拉格朗日方法,为弥补该方法带来的数值耗散,引入了高阶精度紧致格式,在较粗的网格上亦可得到较高精度的导数值．实验结果表明：该算法效果比较真实,速度较快．     

GPU加速的二维地震波场模拟研究

采用交错网格有限差分方法模拟二维地震弹性/粘弹性波场要花费大量的计算时间,为此利用GPU并行处理特点和绘制管道,将计算区域划分为内部区域和PML边界处理区域,整个计算过程由顶点编程和片段编程处理,采用FBO技术实现差分迭代结果的纹理转换。实验结果表明,与CPU实现相比,GPU方法提高了模拟效率,并且随着网格规模的增加,其效率不断提升,可以实现大规模的高效模拟。     

High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

In this paper, we demonstrate the detailed numerical studies of three classical two dimensional detonation waves by solving the two dimensional reactive Euler equations with species with the fifth order WENO-Z finite difference scheme (Borges et al. in J. Comput. Phys. 227:3101?C3211, 2008) with various grid resolutions. To reduce the computational cost and to avoid wave reflection from the artificial computational boundary of a truncated physical domain, we derive an efficient and easily implemented one dimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two dimensional unsteady reactive Euler equation when one of the directions of domain is periodical and inflow/outflow in the other direction. The numerical comparison among characteristic, free stream, extrapolation and PML boundary conditions are conducted for the detonation wave simulations. The accuracy and efficiency of four mentioned boundary conditions are verified against the reference solutions which are obtained from using a large computational domain. Numerical schemes for solving the system of hyperbolic conversation laws with a single-mode sinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated and resolved in the simulations. It is shown that large amplitude of perturbation wave generates more fine scale structures within the detonation waves and the number of cell structures depends on the wave number of sinusoidal perturbation.     

Unified matrix-exponential FDTD formulations for modeling electrically and magnetically dispersive materials

Unified matrix-exponential finite difference time domain (ME-FDTD) formulations are presented for modeling linear multi-term electrically and magnetically dispersive materials. In the proposed formulations, Maxwell?s curl equations and the related dispersive constitutive relations are cast into a set of first-order differential matrix system and the field?s update equations can be extracted directly from the matrix-exponential approximation. The formulations have the advantage of simplicity as it allows modeling different linear dispersive materials in a systematic manner and also can be easily incorporated with the perfectly matched layer (PML) absorbing boundary conditions (ABCs) to model open region problems. Apart from its simplicity, it has been shown that the proposed formulations necessitate less storage requirements as compared with the well-know auxiliary differential equation FDTD (ADE-FDTD) scheme while maintaining the same accuracy performance.     

一类四阶微积分方程的四阶差分格式

针对由吊桥模型而建立的四阶微积分方程,提出了四阶差分格式进行求解．对线性项采用紧格式进行离散,积分项则采用复化辛普森求积公式处理,再结合Newton型迭代法对方程进行求解．给出了差分格式解的存在性和收敛性的证明．数值结果表明格式的精度为O（h4）．     

A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

In this paper, based on the idea of the immersed interface method, a fourth-order compact finite difference scheme is proposed for solving one-dimensional Helmholtz equation with discontinuous coefficient, jump conditions are given at the interface. The Dirichlet boundary condition and the Neumann boundary condition are considered. The Neumann boundary condition is treated with a fourth-order scheme. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.     

Modeling Seismic-Acoustic Fields in Axisymmetric Absorbing Media: the Finite Difference Scheme

The finite difference scheme for modeling seismo-acoustic field propagation in axisymmetric absorbing media excited by an emitter in the borehole fluid (a monopole, a dipole, or a quadrupole) during the acoustic logging, or by an emitter in an elastic medium (a concentrated force, a dipole, or a center of expansion) during the seismic prospecting is presented. The explicit finite difference scheme approximating the equations of the modified Biot’s model, which describes the acoustic wave propagation in an isotropic porous viscoelastic medium saturated with viscous fluid, is proposed.     

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The method of difference potentials was originally proposed by Ryaben??kii and can be interpreted as a generalized discrete version of the method of Calderon??s operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve. Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils??one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation. We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.     

High-order Compact Schemes for Nonlinear Dispersive Waves

High-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge–Kutta scheme are applied to simulate nonlinear dispersive wave propagation problems described the Korteweg-de Vries (KdV)-like equations, which involve a third derivative term. Several examples such as KdV equation, and KdV-Burgers equation are presented and the solutions obtained are compared with some other numerical methods. Computational results demonstrate that high-order compact schemes work very well for problems involving a third derivative term.     

A new fourth-order difference scheme for solving an N-carrier system with Neumann boundary conditions

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods.