Analytical solutions to the three-dimensional radio-nuclide transport equation for computer code verification

A series of analytical solutions to the three-dimensional radio-nuclide transport equation is derived and used to verify the numerical accuracy of the finite-element code NAMSOL. Good agreement is found for grid Peclet numbers up to 8. Above this value the solution becomes unstable as expected from the standard stability criterion.     

Continuous source in tidal flow: A numerical study of the transport equation

A detailed comparative numerical study of a continuous source discharging in a tidal flow has been performed. The advective term in the one-dimensional transport equation consists of a constant freshwater velocity and an isotropic oscillating component. Various finite element solutions are investigated. Compared to the analytical solution of the dynamic steady state concentration, the numerical results for typical estuarine conditions clearly indicate the superiority of the collocation method with Hermite basis functions. An extended Fourier series analysis that accounts for the source condition is developed to explain the numerical behaviour of the different schemes.     

Modeling of solute transport into sub-aqueous sediments

A model for investigating the solute transport into a sub-aqueous sediment bed, under an imposed standing water surface wave, is developed. Under the assumption of Darcy flow in the bed, a model based on a two-dimensional, unsteady advection–diffusion equation is derived; the relative roles of the advective and diffusive transport are characterized by a Peclet number, Pe. Two solutions for the equation are developed. The first is a basic control volume method using the power-law scheme. The second is a smear-free, modified upwind solution for the special case of Pe → ∞. Results, at a given time step, are reported in terms of a laterally averaged solute verse depth profile. The main result of the paper is to demonstrate that the one-dimensional solute concentration verse depth profile is essentially independent of any numerical dissipation present in the solute field predictions. This demonstration is achieved by (i) using an extensive grid refinement study, and (ii) by comparing Pe → ∞ predictions obtained with the basic and smear-free solutions.     

A multiscale finite element method for the solution of transport equations

A multiscale Galerkin finite element scheme based on the residual free bubble function method is proposed to generate stable and accurate solutions for the transport equations namely diffusion-reaction (DR), convection-diffusion (CD) and convection-diffusion-reaction (CDR) equations. These equations show multiscale behavior in reaction or convection dominated situations. The idea is based on the approximation of the definite integral of the interpolation function within the element, instead of the function approximation. The numerical experiments are performed using the bilinear Lagrangian elements. To validate the approach, the numerical results obtained for a benchmark problem are compared with the analytical solution in a wide range of Peclet and Damköhler numbers. The results show that the developed method is capable of generating stable and accurate solutions.     

Global dynamics and zero-diffusion limit of a parabolic–elliptic–parabolic system for ion transport networks

This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained.     

Solution of the advection-dispersion equation with free exit boundary

I investigated the exit boundary condition for the advection-dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non-zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite-domain solution.     

Least-squares spectral method for solving advective population balance problems

The combined CFD-PBE (population balance models) are computationally intensive requiring efficient numerical methods for solving practical problems. In this paper, a high order method is presented based on the least-squares method (LSM) for the solution of a spatial-dependent population balance equation which includes advective processes. Numerical experiments are performed in order to study the behavior of the proposed method for one-dimensional cases using model problems with analytical solutions.     

Wavelet based multiscale scheme for two-dimensional advection–dispersion equation

In this paper, wavelet based adaptive solver is developed for two dimensional advection dominating solute problem which generates sharp concentration front in the solution. In order to handle simultaneously smooth and shock-like behavior, the framework uses finite element discretization followed by wavelets for multiscale decomposition. Daubechies wavelet filter is incorporated to eliminate spurious oscillations at very high Peclet number. The developed solution is compared with the analytical solution to assess the accuracy and robustness. The advantages of the present method over the commonly used methods such as FDM and FEM for solving the problems which show non-physical oscillation in the numerical solution are demonstrated.     

A complete flux scheme for the stationary advection-diffusion-reaction equation

Expressions for the numerical flux of a conservation law of advection-diffusion-reaction type are derived from a local solution of the entire conservation law, including the source term. The resulting complete flux scheme is given for one-dimensional (in Cartesian and spherical coordinates) and two-dimensional model equations. Combined with a finite volume method, the numerical scheme is second order accurate, uniformly in the Peclet numbers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advective transport via error minimization: enforcing constraints of non-negativity and conservation

An operations research method is presented for deriving a conservative, non-negative computational scheme for advective transport. Finite elements in space and time are used to approximate the solution, and the integral of the square of the residual is minimized over the entire spatial domain and over a single temporal element. Negative values are excluded by inequality constraints and conservation is enforced by Lagrange multipliers. The method is then generalized to show how negative values arising in conventional finite-difference methods can be eliminated.     

Eulerian–Lagrangian localized adjoint methods for reactive transport with biodegradation

The microbial degradation of organic contaminants in the subsurface holds significant potential as a mechanism for in-situ remediation strategies. The mathematical models that describe contaminant transport with biodegradation involve a set of advective–diffusive–reactive transport equations. These equations are coupled through the nonlinear reaction terms, which may involve reactions with all of the species and are themselves coupled to growth equations for the subsurface bacterial populations. In this article, we develop Eulerian–Lagrangian localized adjoint methods (ELLAM) to solve these transport equations. ELLAM are formulated to systematically adapt to the changing features of governing partial differential equations. The relative importance of retardation, advection, diffusion, and reaction is directly incorporated into the numerical method by judicious choice of the test functions that appear in the weak form of the governing equation. Different ELLAM schemes for linear variable–coefficient advective–diffusive–reactive transport equations are developed based on different operator splittings. Specific linearization techniques are discussed and are combined with the ELLAM schemes to solve the nonlinear, multispecies transport equations. © 1995 John Wiley & Sons, Inc.     

Anisotropic Burgers equation with an anisotropic perturbation

In this paper we propose an anisotropic Burgers equation with an anisotropic perturbation, which governs many nonlinear dynamic processes in an anisotropic diffusive (viscous) medium. Using conventional transformation methods, we find the analytical solution for the initial-value problem of the Burgers equation in a 2D anisotropic space. Based on the new development of the 2D Laplace’s theorem, we obtain the spatial asymptotic solution as the elements and determinant of the diffusive tensor approach zero. Finally, we also derive the temporal asymptotic solution as time goes to infinity for the 2D anisotropic Burgers system.     

Annular liquid jets in zero gravity

The equations describing the steady-state behavior of long annular liquid jets and liquid membranes in zero gravity are solved analytically as a function of the pressure difference across the jet or membrane, Weber number, and nozzle exit angle. The ranges of the parameters for which the analytical solutions are valid are determined, and analytical solutions of the mass absorption rate are obtained as a function of the Peclet and Weber numbers, nozzle exit angle, pressure difference, and thickness of the annular liquid jet. It is shown that the convergence length of annular liquid jets and liquid membranes increases as the Weber number, nozzle exit angle, and pressure coefficient are increased. It is also shown that the mass absoption rate increases as the nozzle exit angle, pressure coefficient, and Weber number are increased; however, the mass absorption rate decreases as the Peclet number and annular jet initial thickness-to-radius ratio are increased.     

Unique and multiple PHAM series solutions of a class of nonlinear reactive transport model

The purpose of this paper is to visit a class of nonlinear reactive transport model in the case including advective and diffusive transport with the Michaelis-Menten reaction term. We apply the method so-called predictor homotopy analysis method (PHAM) which has been recently proposed to predict multiplicity of solutions of nonlinear BVPs. Consequently two consequential matters are indicated which confirms the authority of PHAM to identify multiple solutions: (i) The Talylor series solutions are improved by the so-called convergence-controller parameter (ii) The possibility of existence of multiple solutions is discovered in some cases for the model.     

A mixed finite difference and boundary element approach to one-dimensional Burgers' equation

This paper presents a mixed method for the numerical solution of the one-dimensional Burgers' equation. This method uses mixed boundary elements in association with finite differences. Two standard problems are used to validated the algorithm. Comparisons are made with some of the existing numerical schemes and analytical solutions. The proposed method performs well.     

Coupled consolidation and contaminant transport model for simulating migration of contaminants through the sediment and a cap

Capping contaminated sediments in waterways is an alternate remediation technique to dredging and is typically much cheaper than dredging. When cap material is placed on top of contaminated sediment, it has both a short-term and long-term hydraulic impact on the underlying sediment. A numerical model of consolidation, based on a nonlinear finite strain theory for a consolidating fine-grained sediment bed was developed. The nonlinear equation of consolidation was solved in a material (or reduced) coordinate using an explicit finite difference numerical scheme. An one-dimensional advection–diffusion equation with sorption and decay was solved on a convective coordinate using a finite volume total variation diminishing (TVD) scheme for the contaminant concentration within the consolidating sediment. The contaminant transport model was coupled with the consolidation model. The time and space varying porosities, permeabilities, and advective velocities computed by the consolidation model were linked to the transport model at the same time level. A number of benchmark tests that are relevant to the consolidation of a fine-grained sediment were designed and tested. The relative contribution of consolidation-induced transient advective velocities on the migration of a contaminant during consolidation was also investigated. The coupled model performance was validated by simulating the transport of hazardous chemicals under consolidation in a confined aquatic disposal (CAD) site in the Lower Duwamish Waterway, Seattle.     

Nonstandard methods for the convective transport equation with nonlinear reactions

A new nonstandard Lagrangian method is constructed for the one-dimensional, transient convective transport equation with nonlinear reaction terms. An “exact” time-stepping scheme is developed with zero local truncation error with respect to time. The scheme is based on nonlocal treatment of nonlinear reactions, and when applied at each spatial grid point gives the new fully discrete numerical method. This approach leads to solutions free from the numerical instabilities that arise because of incorrect modeling of derivatives and nonlinear reaction terms. Algorithms are developed that preserve the properties of the numerical solution in the case of variable velocity fields by using nonuniform spatial grids. Effects of different interpolation techniques are examined and numerical results are presented to demonstrate the performance of the proposed new method. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 467–485, 1998     

Boundary element method and wave equation

In this paper the boundary integral expression for a one-dimensional wave equation with homogeneous boundary conditions is developed. This is done using the time dependent fundamental solution of the corresponding hyperbolic partial differential equation. The integral expression developed is a generalized function with the same form as the well-known D'Alembert formula. The derivatives of the solution and some useful invariants on the characteristics of the partial differential equation are also calculated.The boundary element method is applied to find the numerical solution. The results show excellent agreement with analytical solutions.A multi-step procedure for large time steps which can be used in the boundary element method is also described.In addition, the way in which boundary conditions are introduced during the time dependent process is explained in detail. In the Appendix the main properties of Dirac's delta function and the Heaviside unit step function are described.     

A posteriori error estimation for convection dominated problems on anisotropic meshes

A singularly perturbed convection–diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Smouldering front shapes for steady propagation in a stratified medium

The steady propagation of a smouldering reaction front parallelto the faces of a solid reactive slab has been considered withthe density of the reactive material changing with distancefrom one surface. Such slow steady propagation is known to existin peat bogs where smoulder can take place over many monthsPage et al.(2002, Nature, 420, 61–65). As smoulder progressesit leaves behind a porous matrix through which oxidizer is ableto diffuse to the reaction front. At the front, oxidizer andfuel combine stoichiometrically and the concentration of oxidizeris reduced to zero. In the analysis to be presented the Pecletnumber, based on an assumed constant smoulder speed, is small.The equations and boundary conditions for the oxidizer in theporous region are solved to first order by a complex variablemethod and hodograph transformation. The solution allows theshape of the smouldering front and the oxidizer distributionbehind the front to be determined. Example curves for particularfuel distributions are given. An analysis shows how furtherterms in the expansion of oxidizer concentration in powers ofthe Peclet number may be obtained.