A higher-order eulerian scheme for coupled advection-diffusion transport

A new accurate high-order numerical method is presented for the coupled transport of a passive scalar (concentration) by advection and diffusion. Following the method of characteristics, the pure advection problem is first investigated. Interpolation of the concentration and its first derivative at the foot of the characteristic is carried out with a fifth-degree polynomial. The latter is constructed by using as information the concentration and its first and second derivatives at computational points on current time level t in Eulerian co-ordinates. The first derivative involved in the polynomial is transported by advection along the characteristic towards time level t + Δt in the same way as is the concentration itself. Second derivatives are obtained at the new time level t + Δt by solving a system of linear equations defined only by the concentrations and their derivatives at grid nodes, with the assumption that the third-order derivatives are continuous. The approximation of the method is of sixth order. The results are extended to coupled transport by advection and diffusion. Diffusion of the concentration takes place in parallel with advection along the characteristic. The applicability and precision of the method are demonstrated for the case of a Gaussian initial distribution of concentrations as well as for the case of a steep advancing concentration front. The results of the simulations are compared with analytical solutions and some existing methods.     

Utilization of the method of characteristics to solve accurately two-dimensional transport problems by finite elements

A new finite element method is presented for the solution of two-dimensional transport problems. The method is based on a weighted residual formulation in which the method of characteristics is combined with the finite element method. This is achieved by orienting sides of the space-time elements joining the nodes at subsequent time levels along the characteristics of the pure advection equation associated with the transport problem. The method is capable of solving numerically the advection--diffusion equation without generating oscillations or numerical diffusion for the whole spectrum of dispersion from diffusion only through mixed dispersion to pure convection. The utility and accuracy of the method are demonstrated by a number of examples in two space dimensions and a comparison of the numerical results with the exact solution is presented in one case. A very favourable feature of the method is the capability of solving accurately advection dominated transport problems with very large time steps for which the Courant number is well over one.     

On the use of the reach-back characteristics method for calculation of dispersion

The Holly-Preissmann two-point finite difference scheme (HP method) has been popularly used for solving the advection equation. The key idea of this scheme is to solve the dependent variable (i.e. the concentration for the pollutant transport problem) by the method of characteristics with the use of cubic interpolation on the spatial axis. The interpolating polynomials of higher order are constructed by use of the dependent variable and its derivatives at two adjacent grid points. In this paper a new interpolating technique is introduced for incorporation with the Holly-Preissmann two-point method. The new method is denoted herein as the Holly-Preissmann reach-back method (HPRB) and allows the characteristics to project back several time steps beyond the present time level. Through stability analyses it has been observed that the increase of the reach-back time step numbers for the characteristics indeed reduces the numerical damping and dispersive phenomena. A schematic model has been constructed to demonstrate the merits of this new technique for the calculation of the pure advection and dispersion equations. Numerical experiments and comparisons with analytical solutions which support and demonstrate this new technique are presented.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

A least-squares procedure for the solution of transport problems

In this paper a least-squares formulation associated with a conjugate gradient algorithm is proposed for the solution of transport problems. In this procedure the advection–diffusion equation is first discretized in time using an implicit scheme. At each time step the resulting partial differential equation is replaced by an optimal control problem. This minimization problem involves the minimization of a functional defined via a state equation. This functional is chosen in order to force the numerical solution of the advection–diffusion equation to be equal to the hyperbolic advective part of this equation. The effectiveness of the method is shown through a one-dimensional example involving advective and diffusive transport. No oscillation and high accuracy have been obtained for the entire range of Peclet numbers with a Courant number well in excess of unity.     

Modeling Variable Density Flow and Solute Transport in Porous Medium: 2. Re‐Evaluation of the Salt Dome Flow Problem

Case 5, Level 1 of the international HYDROCOIN groundwater flow modeling project is an example of idealized flow over a salt dome. The groundwater flow is strongly coupled to solute transport since density variations in this example are large (20%).Several independent teams simulated this problem using different models. Results obtained by different codes can be contradictory. We develop a new numerical model based on the mixed hybrid finite elements approximation for flow, which provides a good approximation of the velocity, and the discontinuous finite elements approximation to solve the advection equation, which gives a good approximation of concentration even when the dispersion tensor is very small. We use the new numerical model to simulate the salt dome flow problem.In this paper we study the effect of molecular diffusion and we compare linear and nonlinear dispersion equations. We show the importance of the discretization of the boundary condition on the extent of recirculation and the final salt distribution. We study also the salt dome flow problem with a more realistic dispersion (very small dispersion tensor). Our results are different to prior works with regard to the magnitude of recirculation and the final concentration distribution. In all cases, we obtain recirculation in the lower part of the domain, even for only dispersive fluxes at the boundary. When the dispersion tensor becomes very small, the magnitude of recirculation is small. Swept forward displacement could be reproduced by using finite difference method to compute the dispersive fluxes instead of mixed hybrid finite elements.     

A linearization technique for multi-species transport problems

We study a one-dimensional multi-species system of dispersive-advective contaminant transport equations coupled by nonlinear biological (kinetic reactions) and physical (adsorption) processes. To deal with the nonlinearities and the coupling, and to avoid additional computational costs, we propose a linearization technique based on first-order Taylor’s series expansions. A stabilized finite element in space, combined with an Euler implicit finite difference discretization in time, is used to approximate the dispersive-advective transport problem. Three computational tests are performed with different boundary conditions, retardation factors and kinetic parameters for a nonlinear reactive multi-species transport model. The proposed methodology is shown to be accurate and decrease computational costs in the numerical implementation of nonlinear reactive transport problems.     

High accuracy time and space transform method for advection–diffusion equation in an unbounded domain

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection–diffusion equation in the unbounded domain Rⁿ is converted to one on the bounded domain [?1, 1]ⁿ, and the Laplace transform is applied to eliminate time dependency. The consequent boundary value problem is solved by collocation on Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that TSTM has exponential rate in time and space. Copyright © 2008 John Wiley & Sons, Ltd.     

Accurate and efficient simulation of transport in multidimensional flow

A new numerical method to obtain high‐order approximations of the solution of the linear advection equation in multidimensional problems is presented. The proposed conservative formulation is explicit and based on a single updating step. Piecewise polynomial spatial discretization using Legendre polynomials provides the required spatial accuracy. The updating scheme is built from the functional approximation of the exact solution of the advection equation and a direct evaluation of the resulting integrals. The numerical details for the schemes in one and two spatial dimensions are provided and validated using a set of numerical experiments. Test cases have been oriented to the convergence and the computational efficiency analysis of the schemes. Copyright © 2009 John Wiley & Sons, Ltd.     

基于能度可靠性的结构优化设计方法

基于可靠性的结构优化设计是不确定性结构设计的最合理途径。本文提出了基于能度可靠性的结构优化设计方法，将不确定结构的优化设计描述为：在重量或造价约束下，极小化结构的失效可能度；或对确定的容许失效可能度，极小化结构重量或造价。所提方法和传统的基于随机可靠性的结构优化设计是平行和相似的。由于结构的能度可靠性模型对已知数据的依赖性较低，计算过程较为简便。从而可使结构设计阶段获取数据的难度大大降低，并有效降低计算工作量，且可使模糊信息的处理更为合理。实例计算说明了文中方法是有效和可行的。