A MIXED GREEN ELEMENT FORMULATION FOR THE TRANSIENT BURGERS EQUATION

The transient one-dimensional Burgers equation is solved by a mixed formulation of the Green element method (GEM) which is based essentially on the singular integral theory of the boundary element method (BEM). The GEM employs the fundamental solution of the term with the highest derivative to construct a system of discrete first-order non- linear equations in terms of the primary variable, the velocity, and its spatial derivative which are solved by a two-level generalized and a modified time discretization scheme and by the Newton–Raphson algorithm. We found that the two-level scheme with a weight of 0ċ67 and the modified fully implicit scheme with a weight of 1ċ5 offered some marginal gains in accuracy. Three numerical examples which cover a wide range of flow regimes are used to demonstrate the capabilities of the present formulation. Improvement of the present formulation over an earlier BE formulation which uses a linearized operator of the differential equation is demonstrated. © 1997 by John Wiley & Sons, Ltd.     

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Advances in analytical techniques for polychlorinated dibenzo-p-dioxins, polychlorinated dibenzofurans and dioxin-like PCBs

Analytical techniques for the determination of polychorinated dibenzo-p-dioxins (PCDD), polychlorinated dibenzofurans (PCDF) and dioxin-like PCBs (DLPCB) are reviewed. The focus of the review is on recent advances in methodology and analytical procedures. The paper also reviews toxicology, the development of toxic equivalent factors (TEF) and the determination of toxic equivalent quantity (TEQ) values. Sources, occurrence and temporal trends of PCDD/PCDF are summarized to provide examples of levels and concentration ranges for the methods and techniques reviewed.     

Boundary Integral Procedures for Unsaturated Flow Problems

Abstract. A novel numerical scheme based on the singular integral theory of the boundary element method. (BEM) is presented for the solution of transient unsaturated flow in porous media. The effort in the present paper is directed in facilitating the application of the boundary integral theory to the solution of the highly non-linear equations that govern unsaturated flow. The resulting algorithm known as the Green element method (GEM) presents a robust attractive method in the state-of -the-art application of the boundary element methodology. Three GEM models based on their different methods of handling the non-linear diffusivity, illustrate the suitability and robustness of this approach for solving highly non-linear 1-D and 2-D flows which would have proved cumbersome or too difficult to implement with the classical BEM approach.     

Transient 1D transport equation simulated by a mixed Green element formulation

New discrete element equations or coefficients are derived for the transient 1D diffusion–advection or transport equation based on the Green element replication of the differential equation using linear elements. The Green element method (GEM), which solves the singular boundary integral theory (a Fredholm integral equation of the second kind) on a typical element, gives rise to a banded global coefficient matrix which is amenable to efficient matrix solvers. It is herein derived for the transient 1D transport equation with uniform and non-uniform ambient flow conditions and in which first-order decay of the containment is allowed to take place. Because the GEM implements the singular boundary integral theory within each element at a time, the integrations are carried out in exact fashion, thereby making the application of the boundary integral theory more utilitarian. This system of discrete equations, presented herein for the first time, using linear interpolating functions in the spatial dimensions shows promising stable characteristics for advection-dominant transport. Three numerical examples are used to demonstrate the capabilities of the method. The second-order-correct Crank–Nicolson scheme and the modified fully implicit scheme with a difference weighting value of two give superior solutions in all simulated examples. © 1997 John Wiley & Sons, Ltd.     

A Boundary Element-Finite Element Equation Solutions to Flow in Heterogeneous Porous Media

Abstract. A coupled boundary element-finite element procedure, namely, the Green element method (GEM) is applied to the solution of mass transport in heterogeneous media. An equivalent integral equation of the governing differential equation is obtained by invoking the Green's second identity, and in a typical finite element fashion, the resulting equation is solved on each generic element of the problem domain. What is essentially unique about this procedure is the recognition of the particular advantages and particular features possessed by the two techniques and their effective use for the solution of engineering problems.By utilizing this approach, we observe that the range of applicability of the boundary integral methods is enhanced to cope with problems involving media heterogeneity in a straightforward and realistic manner. The method has been used to investigate problems involving various functional forms of heterogeneity, including head variations in a stream-heterogeneous aquifer interaction and in all these cases encouraging results are obtained without much difficulty.     

Green element method for 2D Helmholtz and convection diffusion problems with variable velocity coefficients

Computation of 2D Helmholtz and transient convection diffusion problems with linear reaction and variable velocity components are implemented with the Green element method (GEM). GEM's fundamental solution which is derived from the diffusion differential operator simplifies the numerical procedure considerably, and together with the Green's second identity, an element to element treatment of the inhomogeneous terms is guaranteed. The reported numerical experiments reveal that the method can be relied on to yield faithful results. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005