共查询到20条相似文献，搜索用时 187 毫秒
1.
《Advances in water resources》2003,26(6):635-647
A kinetic flux vector splitting (KFVS) scheme for shallow water flows based on the collisionless Boltzmann equation is formulated and applied. The scheme is explicit and first order in space and time with stability governed by the Courant condition. The consistency of the KFVS scheme with the shallow water equations is proven using the equivalent differential equations approach. The accuracy and efficiency of the KFVS scheme in modeling complex flow features are compared to those of the Boltzmann Bhatnagar–Gross–Krook (BGK) scheme as well as a Riemann-based scheme. In particular, all schemes are applied to (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) a one-dimensional dam break problem. Additionally, the KFVS, BGK and Riemann schemes are applied to a one-dimensional dam break problem for which laboratory data is available. These test cases reveal that all three schemes provide solutions of comparable accuracy, but the KFVS model is 1.5–2 times faster to execute than the BGK scheme and 2–3 times faster than the Riemann-based scheme. The absence of the collision term from the Boltzmann equation not only makes the mathematical formulation of KFVS easy but also helps elucidate this approach to the novice. The accuracy, efficiency, and simplicity of the KFVS scheme indicate its potential in modeling an array of water resources problems. Due to the scalar nature of the Boltzmann equation, the extension of the KFVS scheme to 2-D surface water flows is straightforward. 相似文献
2.
Alberto Canestrelli Annunziato Siviglia Michael Dumbser Eleuterio F. Toro 《Advances in water resources》2009
This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263–91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103–34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation. 相似文献
3.
The sensitivity of a model output (called a variable) to a parameter can be defined as the partial derivative of the variable with respect to the parameter. When the governing equations are not differentiable with respect to this parameter, problems arise in the numerical solution of the sensitivity equations, such as locally infinite values or instability. An approximate Riemann solver is thus proposed for direct sensitivity calculation for hyperbolic systems of conservation laws in the presence of discontinuous solutions. The proposed approach uses an extra source term in the form of a Dirac function to restore sensitivity balance across the shocks. It is valid for systems such as the Euler equations for gas dynamics or the shallow water equations for free surface flow. The method is first detailed and its application to the shallow water equations is proposed, with some test cases such as dike- or dam-break problems with or without source terms. An application to a two-dimensional flow problem illustrates the superiority of direct sensitivity calculation over the classical empirical approach. 相似文献
4.
A convection-diffusion equation arises from the conservation equations in miscible and immiscible flooding, thermal recovery, and water movement through desiccated soil. When the convection term dominates the diffusion term, the equations are very difficult to solve numerically. Owing to the hyperbolic character assumed for dominating convection, inaccurate, oscillating solutions result. A new solution technique minimizes the oscillations. The differential equation is transformed into a moving coordinate system which eliminates the convection term but makes the boundary location change in time. We illustrate the new method on two one-dimensional problems: the linear convection-diffusion equation and a non-linear diffusion type equation governing water movement through desiccated soil. Transforming the linear convection diffusion equation into a moving coordinate system gives a diffusion equation with time dependent boundary conditions. We apply orthogonal collocation on finite elements with a Crank-Nicholson time discretization. Comparisons are made to schemes using fixed coordinate systems. The equation describing movement of water in dry soil is a highly non-linear diffusion-type equation with coefficients varying over six orders of magnitude. We solve the equation in a coordinate system moving with a time-dependent velocity, which is determined by the location of the largest gradient of the solution. The finite difference technique with a variable grid size is applied, and a modified Crank-Nicholson technique is used for the temporal discretization. Comparisons are made to an exact solution obtained by similarity transformation, and with an ordinary finite difference scheme on a fixed coordinate system. 相似文献
5.
《Advances in water resources》1996,19(3):123-131
Shallow-water flows with supercritical and subcritical subregions often exhibit numerical difficulties because of their associated hydraulic jumps (shock waves), steep layers, and fictitious oscillations. Analogous problems in gas dynamics have led to the recent development of a promising class of Petrov-Galerkin methods specifically designed for hyperbolic/incompletely parabolic systems, and are written in a symmetric conservation form. One of the major difficulties in the application of this class of methods to shallow water problems has been the unavailability of a suitable symmetric form of the governing equations. In the present work, this issue is addressed by introducing the total energy of the water column to motivate a change of variables which symmetrizes the shallow-water conservation system. Then, the one-dimensional case is considered and a time-accurate, streamline-upwind Petrov-Galerkin (SUPG) scheme is developed based on the proposed symmetric form. Numerical results illustrate the method and permit comparison with other schemes. 相似文献
6.
A. C. Fowler 《地球物理与天体物理流体动力学》2013,107(1-4):63-96
Abstract We present a mathematical model for the flow of a partial melt through its solid phase. The model is based on the conservation laws of two-phase flow, which reduce to a generalization of porous flow in a permeable medium, when the solid matrix deforms very slowly. The continuity equation for the melt contains a source term (due to melting), which is determined by the energy equation. In addition, the melt fraction is unknown, and a new equation, representing conservation of pore space, is introduced. This equation may also be thought of as a constitutive law for the melt pressure (which is not lithostatic). The model is non-dimensionalized and simplified. Some simple solutions are considered, and it is suggested that the occurrence of high fluid pressures in the solutions may initiate fractures in the lithosphere, thus providing a starting-up mechanism for magma ascent to the surface. 相似文献
7.
W. Fellin 《Pure and Applied Geophysics》2002,159(7-8):1737-1748
— Godunov's method, a numerical method for solving conservation laws, is applied to nonlinear and inelastic wave propagation in soil. The solution is restricted to the one-dimensional case. An approximate Riemann solver for Godunov's method is presented. The capability of the numerical method is shown by a comparison with the analytical solution of a linear inelastic wave propagation. Finally the behaviour of the nonlinear inelastic soil is described by a hypoplastic constitutive law. 相似文献
8.
从各向同性介质中波场数值模拟的褶积微分算子法出发,推导出了各向异性双相介质中波场传播数值计算的褶积新算法.将常见的二阶微分Biot波动方程用等效的一阶速度—应力双曲方程表示,其中未知的波场向量包括固相和流体的速度分量和应力分量,由此对方程的时间项使用交错网格差分方法计算,而对空间项则采用褶积微分算法进行求解.对各向异性双相介质在单层介质模型和双层介质模型中的波场特征进行了研究.研究的结果显示,在两层介质分界面上当地震波产生反射时能观测到两类纵波和横波,并且在衰减系数大的介质里慢纵波很难见到. 相似文献
9.
A two-phase model for fast geomorphic shallow flows 总被引：1，自引：0，他引：1
The paper introduces a 2D shallow water model based on a two-phase formulation for the analysis of fast geomorphic transients occurring in the context of river morphodynamics.Mass and momentum conservation principles are separately imposed for both phases.The model naturally accounts for non-equilibrium solid transport,since neither instantaneous adaptation hypothesis nor any lag equation is employed to represent sediment dynamics.The hyperbolic character of the proposed model is shown to be preserved independently on the flow conditions.Results from numerical simulations of both 1D and 2D test-cases are compared with literature experimental data and with available numerical solutions. 相似文献
10.
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. 相似文献
11.
Abstract Numerical simulations of internal gravity waves-turbulence are carried out for the inviscid, viscous and forced-dissipative two-dimensional primitive equations using the spectral method. Some of the results are compared with the predictions of the eddy damped quasi-normal Markovian (EDQNM) closure for internal waves of Carnevale and Frederiksen, generalized for periodic boundary conditions and possible random forcing and dissipation. The EDQNM reduces to the Boltzman equation of resonant interaction theory in the continuum space limit and as the forcing and dissipation vanish. However, the limit is singular in the sense that as well as conserving total energy, E, and total cross-correlation between the vorticity and buoyancy fields, C, an additional conservation law, viz. z-momentum, Pz , occurs in the limit. This means that the resonant interaction equilibrium (RIE) solution of the Boltzmann equation differs from the statistical mechanical equilibrium (SME) solution of the EDQNM closure. The statistical stability of the SME and RIE spectra for the primitive equations is tested by integrating the inviscid equations using initial realizations of these spectra with random phases. It is found that E and C are accurately conserved while Pz undergoes large amplitude variations. The approach to equilibrium of initial disequilibrium spectra is monitored by examining the evolution of the entropy. The increase and asymptotic approach to a constant value corresponding to complete chaos is consistent with the behaviour predicted by the EDQNM closure. For the viscous decay and forced-dissipative experiments, the behaviour of the entropy is also consistent with that predicted by the EDQNM closure. There is approximate equipartition of potential and total kinetic energies throughout the integrations from initial conditions having equal potential and total kinetic energies and as well equal vertical and horizontal energies, but as expected, the ratio of horizontal to vertical kinetic energy increases with time to a value greater than unity. With Laplacian viscous dissipation and thermal diffusivity, the statistical steady states produced in the forced-dissipative experiments have k?3 power laws for k≧7. A comparison with the power laws for kinetic energy and passive scalar variance produced in a numerical simulation of the two-dimensional passive scalar problem is also presented. 相似文献
12.
CLAUDIA KERNER 《Geophysical Prospecting》1990,38(2):111-137
Alekseev and Mikhailenko have developed a wavenumber-summation method which combines a finite integral transformation with a finite-difference calculation and involves no approximations other than numerical ones. However, numerical anisotropy causes velocity errors for shear waves which are unacceptable if Poisson's ratios are larger than 0.4 and unless the number of grid points per wavelength is chosen considerably higher than the value generally regarded as sufficient in finite-difference computations. To overcome this limitation in the applicability of the otherwise very powerful modelling scheme, the method is applied to the elastodynamic equations for the velocity vector. Thus, instead of solving a second-order hyperbolic system as in the case of the wave equation, solutions to a first-order hyperbolic system are computed. The finite-difference iteration is performed in a staggered grid. In addition to mastering the numerical difficulties in cases where the Poisson's ratio is unusually high, this approach results in a code which can be used for the modelling of liquid layers. With the new scheme, water reverberations are investigated in terms of normal modes. It is found that for realistic sea-bottom velocities the critical and supercritical cases exist only for P-waves. It means that compressional waves are trapped within the water layer but energy leaks into the substratum through converted shear waves. These leaky compressional normal modes attain properties similar to those of shear normal modes or Pseudo-Love waves. Due to their origin from conversion of dispersed multi-modal compressional waves the shear waves generated at the sea-bottom form a long complex wavetrain. They were found to mask the reflections from the target horizon in an offset-VSP field section. 相似文献
13.
各向异性研究对地下介质精确成像有着重要的意义,在当前计算机硬件迅速发展及宽方位地震数据采集日益普遍的情况下,成像必须考虑介质的各向异性.逆时偏移是基于双程波动方程的较为精确的数值解的成像方法,所以相对于其他地震成像方法,它具有很大的优势,譬如不受反射界面的倾角限制、偏移速度结构合适时能够使回转波及多次波正确成像.在各向同性介质中,可使用标量波方程来模拟波场.而在各向异性介质中,P波和SV波是相互耦合的,即不存在单纯的标量波传播,通常利用能代表耦合波场中P波分量运动学特征的拟声波(qP波)进行偏移成像.本文中,我们推导出了TTI介质下qP波控制方程.该方程可采用显式有限差分格式进行求解.通过声学近似,若沿对称轴方向的剪切波速度为零,对于对称轴方向不变且ε≥δ的模型来说,可得到稳定的数值解.但对于TTI介质来说,由于沿对称轴方向各向异性参数是变化的,声学近似会引起波场传播及数值计算的不稳定.因此,我们提出了正则化有限横波的方法,很好地解决了这一问题.最后,给出了Foothill模型的测试结果及某探区实际资料试算结果,展示了采用这个方程进行复杂TTI模型正演和高质量逆时偏移成像结果,证实了该方法的正确性和实际资料应用中的有效性. 相似文献
14.
We describe two practicable approaches for an efficient computation of seismic traveltimes and amplitudes. The first approach is based on a combined finite‐difference solution of the eikonal equation and the transport equation (the ‘FD approach’). These equations are formulated as hyperbolic conservation laws; the eikonal equation is solved numerically by a third‐order ENO–Godunov scheme for the traveltimes whereas the transport equation is solved by a first‐order upwind scheme for the amplitudes. The schemes are implemented in 2D using polar coordinates. The results are first‐arrival traveltimes and the corresponding amplitudes. The second approach uses ray tracing (the ‘ray approach’) and employs a wavefront construction (WFC) method to calculate the traveltimes. Geometrical spreading factors are then computed from these traveltimes via the ray propagator without the need for dynamic ray tracing or numerical differentiation. With this procedure it is also possible to obtain multivalued traveltimes and the corresponding geometrical spreading factors. Both methods are compared using the Marmousi model. The results show that the FD eikonal traveltimes are highly accurate and perfectly match the WFC traveltimes. The resulting FD amplitudes are smooth and consistent with the geometrical spreading factors obtained from the ray approach. Hence, both approaches can be used for fast and reliable computation of seismic first‐arrival traveltimes and amplitudes in complex models. In addition, the capabilities of the ray approach for computing traveltimes and spreading factors of later arrivals are demonstrated with the help of the Shell benchmark model. 相似文献
15.
《Advances in water resources》1999,22(5):445-460
We develop numerical methods for the simulation of heat and mass transfer in the ground. We use the mixed finite element method for the computation of both the liquid pressure and the temperature, so that the mass is locally and globally well conserved and the computation of the fluid velocity is very accurate. This is quite important since this fluid velocity appears in the first-order term of the equation for the temperature; in turn the partially hyperbolic character of the equation for the temperature is well taken into account since we use the modified method of characteristics for its discretization. 相似文献
16.
17.
在各向异性地震波场中,qP波与qS波常常是耦合在一起的.多分量地震数据处理中一个关键环节就是波型分离(即模式解耦),以纵波成分为主的常规单分量地震数据的成像则需要合理描述标量qP波的传播算子.本文作者曾构建了在运动学上同弹性波动方程等价,动力学上突出标量qP波的伪纯模式波动方程.为了彻底消除qS波残余,本文根据波矢量与qP波偏振矢量之间的偏差,提出从伪纯模式波场提取纯模式标量qP波的方法.数值分析展示了投影偏差算子在波数域和空间域的特征.基于不同复杂程度理论模型的试验结果表明,联合"伪纯模式传播算子"与"投影偏差校正"可为各向异性介质分离模式波场传播过程提供一种简便的描述工具. 相似文献
18.
We propose a wave scattering approach to the problem of deconvolution by the inversion of the reflection seismogram. Rather than using the least-squares approach, we study the full wave solution of the one-dimensional wave equation for deconvolution. Randomness of the reflectivity is not a necessary assumption in this method. Both the reflectivity and the section multiple train can be predicted from the boundary data (the reflection seismogram). This is in contrast to the usual statistical approach in which reflectivity is unpredictable and random, and the section multiple train is the only predictable component of the seismogram. The proposed scattering approach also differs from Claerbout's method based on the Kunetz equation. The coupled first-order hyperbolic wave equations have been obtained from the equation of motion and the law of elasticity. These equations have been transformed in terms of characteristics. A finite-difference numerical scheme for the downward continuation of the free-surface reflection seismogram has been developed. The discrete causal solutions for forward and inverse problems have been obtained. The computer algorithm recursively solves for the pressure and particle velocity response and the impedance log. The method accomplishes deconvolution and impedance log reconstruction. We have tested the method by computer model experiments and obtained satisfactory results using noise-free synthetic data. Further study is recommended for the method's application to real data. 相似文献
19.
地震波传播过程本质上是能量在传播过程中逐步损耗直至殆尽的过程，而在实际应用中，常在无能量损耗假设下，用弹性波动方程或标量波动方程描述它.在哈密顿(Hamilton)体系表述下，地震波传播过程即为一个无限维的哈密顿系统随时间的演化过程.若不计能量损耗，波场演化过程实质上为一个单参数连续的辛变换，因而对应的数值算法应为辛几何算法.本文首先从地震波标量方程出发，给出哈密顿体系下地震波传播的表述，即任意两个时刻的波场是通过辛变换联系起来的.随后，把波场在时间和相空间离散化后，给出了用于波场计算的一些辛格式，如显式辛格式、隐式辛格式和蛙跳辛格式.并进一步讨论了有限差分格式和辛格式的异同.然后，应用显式辛格式和同阶的有限差分方法给出了同一理论速度模型下的波场和Marmousi速度模型下的单炮记录.数值结果表明，辛算法是一类可行的波场模拟的数值算法.在时间步长较小时，有限差分方法是辛算法的一个很好近似.文中的理论和方法，为地震波传播理论及实际应用研究提供了新的途径. 相似文献
20.
常见弹性波动理论的建立是基于介质均匀这一基本假设,实际介质的非均匀性非常普遍.为研究连续介质中波的传播特征,本文从弹性力学中建立弹性波动方程的三个基本方程出发,考虑连续介质弹性参数的空变特征,建立非均匀介质的弹性波动方程,利用Alkhalifah声学近似思想建立位移表征的纵波波动方程,利用本征值问题求解方法建立标量波频率-波数域传播算子,从而建立描述纵波传播的标量波方程,其中波函数为纵波位移的散度,不同于均匀介质标量波方程的波函数为位移势.随后推导含PML边界波动方程差分格式并建立不同模型数值模拟进行数值试算,与均匀假设标量波方程和变密度方程对比证明本方法的准确性和稳定性. 相似文献