An interpolation-corrected modified method of characteristics to solve advection-dispersion equations

Numerical dispersion, numerical oscillation, and peak clipping are common numerical difficulties in solving advection-dispersion equations. The development of numerical approaches that can handle these numerical difficulties with reasonable computational efforts is an ongoing challenge. In this paper, an interpolation-corrected modified method of characteristics (ICMMOC) is proposed for solving advection-dispersion equations. The ICMMOC is an improved version of the modified method of characteristics (MMOC). It uses a high-order (second-order or higher) interpolation scheme to reduce numerical dispersion and an interpolation-correction procedure to eliminate numerical oscillation. A simple peak capturing scheme to overcome the peak clipping problem is also developed in this study. Simulation results show that the ICMMOC is able to overcome the aforementioned numerical difficulties for a large range of grid Peclet numbers.     

Explicit concomitance of implicit method to solve vibration equation

It has been proven that the implicit method used to solve the vibration equation can be transformed into an explicit method,which is called the concomitant explicit method.The constant acceleration met...     

Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method

We extend the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) for solution of the advection-dispersion equation to two dimensions. The method can conserve mass globally and is not limited by restrictions on the size of the grid Peclet or Courant number. Therefore, it is well suited for solution of advection-dominated ground-water solute transport problems. In test problem comparisons with standard finite differences, FVELLAM is able to attain accurate solutions on much coarser space and time grids. On fine grids, the accuracy of the two methods is comparable. A critical aspect of FVELLAM (and all other ELLAMs) is evaluation of the mass storage integral from the preceding time level. In FVELLAM this may be accomplished with either a forward or backtracking approach. The forward tracking approach conserves mass globally and is the preferred approach. The backtracking approach is less computationally intensive, but not globally mass conservative. Boundary terms are systematically represented as integrals in space and time which are evaluated by a common integration scheme in conjunction with forward tracking through time. Unlike the one-dimensional case, local mass conservation cannot be guaranteed, so slight oscillations in concentration can develop, particularly in the vicinity of inflow or outflow boundaries.     

Modeling contaminant transport in homogeneous porous media with fractional advection-dispersion equation

The newly developed Fractional Advection-Dispersion Equation (FADE), which is FADE was extended and used in this paper for modelling adsorbing contaminant transport by adding an adsorbing term. A parameter estimation method and its corresponding FORTRAN based program named FADEMain were developed on the basis of Nonlinear Least Square Algorithm and the analytical solution for one-dimensional FADE under the conditions of step input and steady state flow. Data sets of adsorbing contaminants Cd and NH4+-N transport in short homogeneous soil columns and conservative solute NaCI transport in a long homogeneous soil column, respectively were used to estimate the transport parameters both by FADEMain and the advection-dispersion equation (ADE) based program CXTFIT2.1. Results indicated that the concentration simulated by FADE agreed well with the measured data. Compared to the ADE model, FADE can provide better simulation for the concentration in the initial lower concentration part and the late higher concentration part of the breakthrough curves for both adsorbing contaminants. The dispersion coefficients for ADE were from 0.13 to 7.06 cm2/min, while the dispersion coefficients for FADE ranged from 0.119 to 3.05 cm1.856/min for NaCI transport in the long homogeneous soil column. We found that the dispersion coefficient of FADE increased with the transport distance, and the relationship between them can be quantified with an exponential function. Less scale-dependent was also found for the dispersion coefficient of FADE with respect to ADE.     

Comparative error analysis in finite element formulations of the advection-dispersion equation

The behaviour of numerical solutions of the one-dimensional advection-dispersion equation is investigated and comparisons between the consistent and the lumped formulations of Galerkin finite element schemes are made. Well-known criteria for the control of accuracy in the lumped (finite difference) formulation are reviewed. It is found that, because the numerical error produced by the consistent formulation is generally less than that produced by the lumped formulation, these criteria can also be used for the control of numerical dispersion in the consistent formulation. However, because the error in both types of solutions decreases in time when the discretization is invariant, the criteria can be relaxed with advancing simulation time. For the consistent formulation it is found that beyond some initial time period, the numerical error depends only on the temporal discretization. This suggests that constant accuracy can be maintained throughout the simulation period while allowing the time step length to grow.     

Exact analytical solutions for two-dimensional advection-dispersion equation in cylindrical coordinates subject to third-type inlet boundary condition

Exact analytical solutions for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.     

A finite element method for the diffusion-convection equation with constant coefficients

Existing numerical methods for the solution of the diffusion-convection equation are unsatisfactory for convection dominated flow problems. A new finite element method incorporating the method of characteristics for the solution of the diffusion-convection equation with constant coefficients in one spatial dimensions is derived. This method is capable of solving diffusion-convection equation without any of the difficulties encountered in the existing numerical methods for the whole spectrum of dispersion from pure diffusion, through mixed dispersion, to pure convection. Several examples for the one-dimensional case are solved and results are compared with the exact solutions. The generalization of the method to variable coefficients and to the diffusion-convection equation in two space dimensions are discussed.     

A comparative method for various approaches to the isothermal equation of state

Various three-parameter approximations of the isothermal equation of state of matter can be systematized and, in consequence, mutually compared by using the uniquely associated relationships between first and second pressure derivatives of the bulk modulus of any substance at zero pressure. Thus quantitative or, at least qualitative information on the capability of such approximations for volumetric contractions down to about 0.5 can be obtained directly. This comparative method will be discussed with the aid of 19 approximations considered in high-pressure physics and geophysics. Its application to the Grüneisen parameter analysis yields a number of additional practical approximations.     

A new method for the analytical solution of a degenerate diffusion equation

Certain nonlinear diffusion equations of degenerate parabolic type display a finite speed of propagation of disturbances. This mathematical behavior can be used to describe a wide range of nonlinear phenomena such as the penetration distance of a thermal layer, the boundary of a reaction zone, or a wetting front in unsaturated soil moisture flow. However, there are two main difficulties in obtaining solutions to problems of this class. One is that the location of the interface is not known a priori and must be discovered during the analysis. The other is the fact that the differential equation is singular in the neighborhood of the interface. The solution technique developed and presented in this work overcomes these difficulties by extracting a local solution of the differential equation in the neighborhood of the diffusing front. One profound result is the discovery that the velocity of the front is entirely controlled by the first term of the spectral series expansion. Also, by capturing the critical behavior of the solution in the region of the singularity and incorporating the behavior as a dominant factor, the series expansion is provided a means for very rapid convergence. The versatility of the solution technique is demonstrated by solving various boundary value problems covering a broad range of interest and the solutions are tested against previously published results.     

On a spectral method of solving the Stokes equation

A method of solving the Stokes equation for a spherical mantle model by expansion in spherical harmonics was developed by Hager and O’Connell [1979]. However, this method is applicable only if the viscosity depends solely on depth. In this case, the Stokes equation reduces to a system of independent equations for each harmonic. Given lateral variations in viscosity, the Stokes equation contains terms in the form of products of harmonics, which invalidates all advantages of harmonic expansion. Zhang and Christensen [1993] developed a perturbation method for the case when terms containing products of lateral viscosity variations are small. These terms are first calculated from the preceding iteration and are then expanded in a series of harmonic functions. As a result, equations for harmonics remain independent. An evident advantage of the spectral method is the simplicity of the technique of incorporating the self-gravitation and compressibility effects. Moreover, this method partially eliminates difficulties related to the singularities at poles. As yet, it has not been applied in practice, possibly because the equations presented in [Zhang and Christensen, 1993] contain misprints that have not been elucidated in the literature. In the present work, a system of equations is derived for the spectral-iterative method of solving the Stokes equation and the errata present in formulas of Zhang and Christensen [1993] and significantly affecting results of calculations are analyzed.     

居里等温面深度的探测方法

地壳居里等温面的研究对确定深部磁体结构和了解深部热状态及结构有重要意义.本文对探测居里等温面深度的方法进行了科学地总结,分类和评估,并主要介绍了各种方法的基本原理.较直接的探测居里等温面的方法是根据大地热流值,地温\|深度关系或岩石矿物实验来确定,但是这种方法在实际中并不常用.最常用的方法是应用航磁和卫星磁数据.根据所研究的目标的不同,磁法又可以被分为两大类：单体磁异常法和组合磁异常法.这两大类下又可以有各种不同的应用技术,其中功率谱分析方法最为流行.虽然没有任何一种方法是完美的,针对组合磁异常的统计功率谱分析方法被认为是较成熟的一项技术,并得到了很多的应用及讨论.组合磁异常的功率谱分析方法的理论基础是Spector\|Grant 统计模型,这个统计模型很适用于对区域磁异常组合的分析.为了提高分辨率,无论单体磁异常法或组合磁异常法都需要考虑很大的一片研究区域,这样的结果会降低所获得的居里等温面的横向分辨率,由此所获得的居里等温面深度通常只是某一区域平均居里点深度的一级近似.一项比较特别的技术是Mayhew的最初为分析卫星磁数据而建立起来的等层模型方法.这一方法的基本原理是,根据一些校准点,将一等厚度层上所观测到的不同的磁场强度转换为等磁场强度的深度差异.这一技术与功率谱分析方法的一个重要差别在于它不需要对磁体顶部和底部所引起的功率谱进行区分,而这种区分是功率谱分析方法中所必需的.由于对居里等温面深度的探测是一种反演技术并包含了反演中的非唯一性和数学上的不稳定性,在各种方法的应用中应采取谨慎的态度.近年来,自相似和分形理论的应用为这一领域的研究提供了很多崭新的研究机会和科学思想.自相似和分形模型是对Spector\|Grant统计模型的完善和发展,因为自相似模型考虑了更真实的磁场强度差异的分形分布.另外,这一新理论也揭示了以前很多种方法中的局限性.     

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.     

The generalized kinetic equation as a model for the nonlinear transfer in third-generation wave models

An alternative model for the nonlinear interaction term S_nl in spectral wave models, the so called generalized kinetic equation (Janssen J Phys Oceanogr 33(4):863–884, 2003; Annenkov and Shrira J Fluid Mech 561:181–207, 2006b; Gramstad and Stiassnie J Fluid Mech 718:280–303, 2013), is discussed and implemented in the third generation wave model WAVEWATCH-III. The generalized kinetic equation includes the effects of near-resonant nonlinear interactions, and is therefore able, in theory, to describe faster nonlinear evolution than the existing forms of S_nl which are based on the standard Hasselmann kinetic equation (Hasselmann J Fluid Mech 12:481–500, 1962). Numerical simulations with WAVEWATCH have been carried out to thoroughly test the performance of the new form of S_nl, and to compare it to the existing models for S_nl in WAVEWATCH; the DIA and WRT. Some differences between the different models for S_nl are observed. As expected, the DIA is shown to perform less well compared to the exact terms in certain situations, in particular for narrow wave spectra. Also for the case of turning wind significant differences between the different models are observed. Nevertheless, different from the case of unidirectional waves where the generalized kinetic equation represents a obvious improvement to the standard forms of S_nl (Gramstad and Stiassnie 2013), the differences seems to be less pronounced for the more realistic cases considered in this paper.     

A comparison of numerical schemes to solve the magnetic induction eigenvalue problem in a spherical geometry

We compare various methods of solving the magnetic induction eigenvalue problem in a sphere, each using toroidal–poloidal decomposition and spherical harmonics, but with a different radial discretisation. In the case of quiescent flow where only diffusion acts upon the magnetic field, we benchmark numerical convergence against the analytic decay rates, and find that a Galerkin scheme based on Chebyshev polynomials with an associated projection chosen such that the diffusion operator is self-adjoint, exhibits the fastest convergence of the schemes described. The importance of the speed of convergence becomes heightened with the introduction of a non-quiescent flow because of the reduction in the magnetic field length scales. We find that sufficiently converged solutions are generally difficult to locate unless we use the optimal Galerkin scheme.     

A general mixing equation with applications to Icelandic basalts

The mixing equation applied by Vollmer [1] to Pb and Sr isotope ratios is shown to be a general equation applicable to consideration of element and isotope ratios. The mixing equation is hyperbolic and has the form:Ax + Bxy + Cy + D = 0where the coefficients are dependent on the type of plot considered: i.e. ratio-ratio, ratio-element, or element-element. Careful use of this equation permits testing whether mixing is a viable process, places constraints on end member compositions, allows distinction between mixing of sources and mixing of magmas, and should allow distinction between recent mixing and long-term evolution of sources.The available chemical data for postglacial basalts from Iceland and along the Reykjanes Ridge are not consistent with either mixing of magmas or simple mixing of an enriched ocean island source with a depleted ocean ridge source. If the available analyses for basalts are representative of the source regions, the data are consistent with at least two models neither of which can be properly tested with the available data.(1) There are two separate mixing trends: one beneath Iceland with the alkali basalt source and a depleted Iceland source as end members; the second along the Reykjanes Ridge with a heterogeneous ocean ridge basalt source and a source similar to that for intermediate basalts on Iceland as end members. The depleted Iceland source and the depleted ocean ridge source are not the same.(2) The chemistry of the basalts is not determined by mixing. Instead the basalts are derived from a multiplicity of sources with a similar history which have been isolated for hundreds of millions of years.     

A method for the study of the zonal kinetic energy balance in the atmosphere

Summary The atmospheric balance of the kinetic energy of the zonally averaged zonal motion is investigated from five years of daily data at 800 stations for the northern hemisphere. The basic equation for such energy is used, together with the simplifying assumption that the frictional destruction is due in the main to stresses acting across horizontal surfaces, being thus presumably related to the vertical shear of the mean zonal wind, although no further details are needed in the analysis. The five-year averages of various terms as well as their seasonal means appear to give reasonable results.The research reported in this paper was sponsored by the U.S. National Science Foundation under Grant No. GA-1310X.     

Boundary integration equation method to scattering of plane SH-waves

In this paper, several approaches used in the boundary integration equation method have been discussed firstly. The comparison among solutions of equation in frequency domain established by the above approaches indicates the features and the validity respectively. Moreover, the application of boundary integration equation method using series solution to scattering of plane SH seismic waves in homogeneous, isotropic medium has been discussed chiefly. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,15, 331–338, 1993.     

基于牛顿迭代法求解群速度的地震波射线追踪模拟

地下介质中普遍存在着各向异性,当前基于各向异性的地震波射线追踪多是在弱各向异性介质中进行且采用群速度近似表示方法,这些近似方法在强各项异性介质中会导致很大误差而无法真正模拟地震波的传播规律.根据地下普遍存在各向异性的事实和地震波基本传播规律,提出利用牛顿迭代法高效求解群速度,基于Paraview平台 自动化构建三维地质...     

改进的一阶弹性波方程交错网格高阶差分解法

本文对董良国、马在田等人发表在<地球物理学报>上的一阶弹性波方程交错网格高阶差分解法一文进行了差分格式的改进.发现原文对一阶空间导数采用2N阶差分精度,而对高阶导数却采用简单的中心差分,即对同一方程中的各阶导数的处理存在有的采用高阶格式、有的采用低阶格式的不一致性问题,本文的改进点正是消除了这种不一致性,建立了时间和空间均为四阶精度的差分格式.该方法具有编程简便、易于计算机实现和精度较高的特点.     

A finite analytic method for solving the 2-D time-dependent advection–diffusion equation with time-invariant coefficients

Difficulty in solving the transient advection–diffusion equation (ADE) stems from the relationship between the advection derivatives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative terms but rather solves the equation analytically in the space–time domain. The method is computationally efficient and numerically accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solution methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based finite analytic method for solving the two-dimensional steady-state advection–diffusion equation. Water Resour Res 38 (7), 10.1029/2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method of characteristics, a random walk particle tracking method, and an Eulerian–Lagrangian Localized Adjoint Method using various degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more efficient while producing similar or better accuracy than the other methods.