Akpofure E. Taigbenu

Green element simulations of the transient nonlinear unsaturated flow equation

This paper presents numerical solutions to transient one-dimensional (1-D) and 2-D nonlinear flows in unsaturated media which are based on the Green element method (GEM). GEM is a novel numerical scheme based essentially on the singular integral theory of the boundary element method (BEM) but which implements the theory in an element-by-element fashion so that when the discrete equations from all the elements are assembled the resultant global coefficient matrix is banded and easier to invert, in contrast to the fully populated global coefficient matrix commonly associated with conventional BEM. Because GEM implements the singular integral theory of BEM within each element, the elemental integrals of the theory are evaluated analytically, thereby enhancing the accuracy of the method, and medium properties that vary spatially and temporally as a result of their dependence on the flow are readily incorporated into the theory. The transient nonlinear flow problem in unsaturated porous medium with soil constitutive relations that are of practical interest is highly nonlinear and had not been amenable to the boundary element theory, but here, for the first time, such problems are successfully solved by that theory using the Green element approach which incorporates a Picard nonlinear solution algorithm. Two Green element models for transient 1-D and 2-D unsaturated flows are developed and successfully tested on four examples of infiltration flow problems whose soil constitutive relations cover most of those reported in the literature.

Green's function-based integral approaches to nonlinear transient boundary-value problems (II)

Abstract Two Greens function-based numerical formulations are used to solve the time-dependent nonlinear heat conduction (diffusion) equation. These formulations, which are an extension of the first paper, utilize two fundamental solutions and the Greens second identity to achieve integral replications of the governing partial differential equation. The integral equations thus derived are discretized in space and time and aggregated in a finite element sense to give a system of nonlinear discrete equations that are solved by the Newton–Raphson algorithm. The mathematical simplicity of the Greens function of the first formulation facilitates its numerical implementation. The performance of the formulations is assessed by comparing their results with available numerical and analytical solutions. In all cases satisfactory and physically realistic results are obtained.

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.

A flux-correct Green element model of quasi three-dimensional multiaquifer flow.

Transient flow in multiply layered aquifers, separated by connecting layers of aquitards which provide hydraulic interactions between the aquifers, is solved by the Green element method (GEM) in a manner that reveals one of its strengths of being able to correctly model the leakage flux without resorting to adjusting the grid representing the one-dimensional (1-D) flow in the aquitards, as done in the finite element method (FEM). The hydraulic approach of approximating the flow is adopted so that flow in the aquifer takes place in two dimensions and that in the aquitards takes place in the one-dimensional vertical direction. The 1-D Green element (GE) model earlier developed for transient diffusion and referred to as the transient GE (TGE) formulation [Taigbenu and Onyejekwe, 1999] is used in modeling the flow in the aquitards, while the 2-D GE model developed for linear and nonlinear transient diffusion [Taigbenu and Onyejekwe, 1998] is used for calculating the flows in the confined and unconfined aquifers. Both models are coupled to solve regional flow problems in multiaquifer systems of arbitrary geometry which receive point and distributed recharge of arbitrary strengths. The solution procedure, which is iterative, provides information on the hydraulic heads and fluxes in the aquifers and aquitards at various specified times. Because GEM is founded on the singular integral theory, singularities that arise from water abstractions at wells (point recharge) are naturally captured in the singular Greens function, thereby making it possible to use a more coarse grid for problems in which there exist active wells. Furthermore, the implementation procedure of GEM achieves sparsity of the coefficient matrix so that less amount of computing resources is required for its decomposition. The superiority of the current approach over FEM in predicting the leakage flux through the aquitards and achieving comparable accuracy for well problems with coarser grid is demonstrated.

Three Green element models for the diffusion–advection equation and their stability characteristics

Abstract Solutions of the transient 1-D diffusion–advection equation by three models of the Green element method (GEM) and their stability characteristics are presented. GEM is a novel approach of implementing the singular boundary integral theory so that computational efficiency is enhanced, and the theory is made more versatile. The first model, denoted as the quasi-steady Green element (QSGE) model, employs the Green’s function of the Laplacian operator in deriving its integral representation, while the second, denoted the TGE model, uses the Green’s function of the transient diffusion differential operator, and the third, denoted the ADGE model, uses the Green’s function of the diffusion–advection differential operator. The first model, which had earlier been presented, is herein compared to the other two models. Three numerical examples are used to compare the accuracies of the three models. It is observed that incorporating the Crank–Nicholson scheme into the first model not only gives optimal results of the three models, but it more readily accommodates transport with nonuniform flow velocity field and first-order rate of decay of the pollutant. Further, the mathematical simplicity of the Green’s function of the first model is an added advantage which enhances computational efficiency. The numerical stability characteristics of these models are evaluated by examining their propagation of the amplitudes and speeds of Fourier wave components in relation to their corresponding theoretical values. The results from the stability analysis confirms the superiority of the QSGE model with the Crank–Nicholson scheme.

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.

Numerical stability characteristics of a Hermitian Green element model for the transport equation

The cubic Hermitian interpolation functions are incorporated into the Green element method (GEM) for the solution of the transient 1-D transport equation, and the stability characteristics of the numerical model are evaluated by examining its propagation characteristics of Fourier wave components vis-a-vis their theoretical values. The results of the Hermitian model are compared to that which approximates the primary variable by linear interpolation functions (linear model), and we observe that the former gives more superior stability characteristics.

Green's function-based integral approaches to linear transient boundary-value problems and their stability characteristics (I)

Abstract Two Greens function-based formulations are applied to the governing differential equation which describes unsteady heat or mass transport in an isotropic homogeneous 1-D domain. In this first part of a two series of papers, the linear form of the differential equation is addressed. The first formulation, herein denoted the quasi-steady Green element (QSGE) formulation, uses the Laplace differential operator as auxiliary equation to obtain the singular integral representation of the governing equation, while the second, denoted the transient Green element (TGE), uses the transient heat equation as auxiliary equation. The mathematical simplicity of the Greens function of the first formulation enhances the ease of solution of the integral equations and the resultant discrete equations. From the point of computational convenience, therefore, the first formulation is preferred. The stability characteristics of the two formulations are evaluated by examining how they propagate various Fourier harmonics in speed and amplitude. We found that both formulations correctly reproduce the theoretical speed of the harmonics, but fail to propagate the amplitude of the small harmonics correctly for Courant value of about unity. The QSGE formulation with difference weighting values between 0.67 and 0.75, and the TGE formulation provide optimal performance in numerical stability.

Green element calculations of nonlinear heat conduction with a time-dependent fundamental solution

Abstract Calculations of nonlinear transient heat conduction are carried out with a Green element formulation that incorporates the time-dependent Greens function derived from the diffusion differential operator. This formulation is different from the one that uses the logarithmic fundamental solution, and offers another viable approach at solving nonlinear heat transfer problems. Applying the formulation in 2D spatial domains, the integral equation arising from applying the singular integral theory is implemented from element to element with linear interpolation in space and time for the temperature field. The nonlinear discretized equations are solved by the Picard and Newton–Raphson algorithms with good convergence being achieved for all thermo-elastic relations examined, and the latter algorithm exhibiting slightly better convergence characteristics. Comparison of the current Green element formulation with the previous one that uses the logarithmic Greens function indicates that comparable accuracy are achievable from both formulations with the latter having an edge in terms of simplicity of formulation.

Features of a time-dependent fundamental solution in the Green element method

Abstract New sets of element coefficients are derived for the Green element method (GEM) which incorporates the time-dependent fundamental solution of the linear diffusion differential operator in two spatial dimensions. These coefficients are obtained for both rectangular and triangular grids when Green element calculations are carried out for heat transfer and contaminant transport problems. Similar coefficients had earlier been derived for these two problems in one spatial dimensions [Appl. Math. Model. 22 (1998) 687; Eng. Anal. Boundary Elements 23 (1999) 577]. The flexibility offered by GEM, that allows the integral representation of the differential operator to be evaluated strictly within a typical element, is exploited by switching the order of integration in time and space to achieve, to a large extent, exact expressions of the element coefficients. In this way the accuracy of the numerical calculations is preserved. Comparison of the current formulation with an earlier one that incorporates the logarithmic fundamental solution indicates that the current formulation does better than the previous one for the heat transfer problem, but the reverse is the case for the contaminant transport problem. As with the 1-D formulation, the current 2-D formulation produces better solutions for the transport problem using larger time step––a numerical feature previously established by the numerical stability analysis [Eng. Anal. Boundary Elements 23 (1999) 577].