Process splitting of the boundary conditions for the advection–dispersion equation

Rational strategies are considered for the specification of the intermediate boundary condition at an inflow boundary where process splitting (fractional steps) is adopted in solving the advection–dispersion equation. Three lowest-order methods are initially considered and evaluation is based on comparisons with an analytical solution. For flow and dispersion parameter ranges typical of rivers and estuaries, the given boundary condition for the complete advection–dispersion equation at the end of the complete time step provides a satisfactory estimate of the intermediate boundary value. This was further confirmed by the development and evaluation of two higher-order methods. These required non-centred discrete approximations for spatial derivatives, which offset any special advantages from the higher truncation error order.     

Finite difference methods for solving the two-dimensional advection–diffusion equation

Using weighted discretization with the modified equivalent partial differential equation approach, several accurate finite difference methods are developed to solve the two-dimensional advection–diffusion equation following the success of its application to the one-dimensional case. These new methods are compared with the conventional finite difference methods in terms of stability and accuracy. The new methods are more accurate and often more stable than the conventional schemes.     

A multiscale control volume finite element method for advection–diffusion equations

We present a new stabilized method for advection–diffusion equations, which combines a control volume FEM formulation of the governing equations with a novel multiscale approximation of the total flux. The latter incorporates information about the exact solution that cannot be represented on the mesh. To define this flux, we solve the governing equations along suitable mesh segments under the assumption that the flux varies linearly along these segments. This procedure yields second‐order accurate fluxes on the edges of the mesh. Then, we use curl‐conforming elements of the same order to lift these edge fluxes into the mesh elements. In so doing, we obtain a stabilized control volume FEM formulation that is second‐order accurate and does not require mesh‐dependent stabilization parameters. Numerical convergence studies on uniform and nonuniform grids along with several standard advection tests illustrate the computational properties of the new method. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.     

Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

A velocity–vorticity formulation of the Navier–Stokes equations is presented as an alternative to the primitive variables approach. The velocity components and the vorticity are solved for in a fully coupled manner using a Newton method. No artificial viscosity is required in this formulation. The pressure is updated by a method allowing natural imposition of boundary conditions. Incompressible and subsonic results are presented for two-dimensional laminar internal flows up to high Reynolds numbers.     

Numerical modelling of advection–dispersion equation in a stretched curvilinear grid using the QUICKEST scheme

A stretched version of the QUICKEST scheme for solutions of the advection–dispersion equation is presented. The scheme is accurate for large degrees of stretching, so that it can be used where large gradients are present, e.g. for the calculation of sediment in suspension close to the bed. The scheme is tested for various cases of sediment advection and dispersion in one and two dimensions. © 1998 John Wiley & Sons, Ltd.     

Augmented upwind numerical schemes for the groundwater transport advection‐dispersion equation with local operators

When solute transport is advection‐dominated, the advection‐dispersion equation approximates to a hyperbolic‐type partial differential equation, and finite difference and finite element numerical approximation methods become prone to artificial oscillations. The upwind scheme serves to correct these responses to produce a more realistic solution. The upwind scheme is reviewed and then applied to the advection‐dispersion equation with local operators for the first‐order upwinding numerical approximation scheme. The traditional explicit and implicit schemes, as well as the Crank‐Nicolson scheme, are developed and analyzed for numerical stability to form a comparison base. Two new numerical approximation schemes are then proposed, namely, upwind–Crank‐Nicolson scheme, where only for the advection term is applied, and weighted upwind‐downwind scheme. These newly developed schemes are analyzed for numerical stability and compared to the traditional schemes. It was found that an upwind–Crank‐Nicolson scheme is appropriate if the Crank‐Nicolson scheme is only applied to the advection term of the advection‐dispersion equation. Furthermore, the proposed explicit weighted upwind‐downwind finite difference numerical scheme is an improvement on the traditional explicit first‐order upwind scheme, whereas the implicit weighted first‐order upwind‐downwind finite difference numerical scheme is stable under all assumptions when the appropriate weighting factor (θ) is assigned.     

Refining the submesh strategy in the two‐level finite element method: application to the advection–diffusion equation

We introduce a new submesh strategy for the two‐level finite element method. The numerical results show that the new submesh is able to better capture the boundary layer which is caused by the choice of bubble functions. The effect of an improved approximation of the residual free bubbles is studied for the advective–diffusive equation. Copyright © 2002 John Wiley & Sons, Ltd.     

A stabilized formulation for the advection–diffusion equation using the Generalized Finite Element Method

This paper presents a stable formulation for the advection–diffusion equation based on the Generalized (or eXtended) Finite Element Method, GFEM (or X‐FEM). Using enrichment functions that represent the exponential character of the exact solution, smooth numerical solutions are obtained for problems with steep gradients and high Péclet numbers in one‐ and two‐dimensions. In contrast with traditional stabilized methods that require the construction of stability parameters and stabilization terms, the present work avoids numerical instabilities by improving the classical Galerkin solution with enrichment functions (that need not be polynomials) using GFEM, which is an instance of the partition of unity framework. This work also presents a strategy for constructing enrichment functions for problems involving complex geometries by employing a global–local‐type approach. Representative numerical results are presented to illustrate the performance of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Conjugate gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patterns

Preconditioning techniques based on incomplete Gaussian elimination for large, sparse, non-symmetric matrix systems are described. A certain level of fill-in may be specified in the incomplete factorizations. All methods considered may be applied to matrices with arbitrary sparsity patterns, for instance those associated with the general preprocessor algorithms or adaptive mesh techniques. The preconditioners have been combined with five conjugate gradient-like methods and tested on finite element discretized scalar convection-diffusion equations in 2D and 3D. It is found from numerical experiments that an amount of fill-in corresponding to about 50% of the number of original non-zero matrix entries is the optimal choice for this class of preconditioners. The preconditioners show almost no sensitivity to grid distortion. In problems with significantly variable coefficients or anisotropy the preconditioners stabilize the basic iterative schemes in addition to reducing the computational work substantially, mostly by more than 90%. The modified preconditioning technique, where fill-in is added on the main diagonal, performs in general better than the standard incomplete LU factorization, but is inferior to the latter in 3D problems and for matrix systems with complicated sparsity patterns.     

Iterative solution techniques for the stokes and Navier-Stokes equations

The linear system arising from a Lagrange-Galerkin mixed finite element approximation of the Navier–Stokes and continuity equations is symmetric indefinite and has the same block structure as a system arising from a mixed finite element discretization of a Stokes problem. This paper considers the iterative solution of such a system, comparing the performance of the one-level preconditioned conjugate residual method for indefinite matrices with that of a more traditional two-level pressure correction approach. Asymptotic estimates for the amount of work involved in each method are given together with the results of related numerical experiments.     

A finite element solution of the time-dependent incompressible Navier–Stokes equations using a modified velocity correction method

A finite element solution of the two-dimensional incompressible Navier–Stokes equations has been developed. The present method is a modified velocity correction approach. First an intermediate velocity is calculated, and then this is corrected by the pressure gradient which is the solution of a Poisson equation derived from the continuity equation. The novelty, in this paper, is that a second-order Runge–Kutta method for time integration has been used. Discretization in space is carried out by the Galerkin weighted residual method. The solution is in terms of primitive variables, which are approximated by polynomial basis functions defined on three-noded, isoparametric triangular elements. To demonstrate the present method, two examples are provided. Results from the first example, the driven cavity flow problem, are compared with previous works. Results from the second example, uniform flow past a cylinder, are compared with experimental data.     

Finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements: plane Poiseuille flow

This paper presents a comprehensive review of the numerical techniques used during the past half century and their accuracy in hydrodynamic stability analysis of plane parallel flows. The paper also describes a finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements. A stability analysis technique is performed by imposing an infinitesimal perturbation to the laminar base flow to determine the thresholds of neutral instabilities or the growth rate of the perturbation for any Reynolds and wave numbers. Validation of the present numerical technique is performed for plane Poiseuille flow. The numerical results, obtained with uniform and nonuniform meshes, show excellent agreement with the most accurate results available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Computation of the stabilization parameter for the finite element solution of advective–diffusive problems

In a previous paper a general procedure for deriving stabilized finite element schemes for advective type problems based on invoking higher order balance laws over finite size domains was presented. This provides an expression for the element stabilization parameter in terms of the solution residual and its first derivatives in a kind of iterative or adaptative manner. Details of the application of this procedure to 1D and 2D advective–diffusive problems are given. Some examples of applications showing the potential of the new approach are presented. © 1997 John Wiley & Sons, Ltd.     

Comparison of Galerkin and control volume finite element for advection–diffusion problems

The control volume finite element method (CVFEM) was developed to combine the local numerical conservation property of control volume methods with the unstructured grid and generality of finite element methods (FEMs). Most implementations of CVFEM include mass‐lumping and upwinding techniques typical of control volume schemes. In this work we compare, via numerical error analysis, CVFEM and FEM utilizing consistent and lumped mass implementations, and stabilized Petrov–Galerkin streamline upwind schemes in the context of advection–diffusion processes. For this type of problem, we find no apparent advantage to the local numerical conservation aspect of CVFEM as compared to FEM. The stabilized schemes improve accuracy and degree of positivity on coarse grids, and also reduce iteration counts for advection‐dominated problems. Published in 2005 by John Wiley & Sons, Ltd.     

Arnoldi and Crank–Nicolson methods for integration in time of the transport equation

A comparison is made between the Arnoldi reduction method and the Crank–Nicolson method for the integration in time of the advection–diffusion equation. This equation is first discretized in space by the classic finite element (FE) approach, leading to an unsymmetric first‐order differential system, which is then solved by the aforementioned methods. Arnoldi reduces the native FE equations to a much smaller set to be efficiently integrated in the Arnoldi vector space by the Crank–Nicolson scheme, with the solution recovered back by a standard Rayleigh–Ritz procedure. Crank–Nicolson implements a time marching scheme directly on the original first‐order differential system. The computational performance of both methods is investigated in two‐ and three‐dimensional sample problems with a size up 30 000. The results show that in advection‐dominated problems less then 100 Arnoldi vectors generally suffice to give results with a 10⁻³–10⁻⁴ difference relative to the direct Crank–Nicolson solution. However, while the CPU time with the Crank–Nicolson starts from zero and increases linearly with the number of time steps used in the simulation, the Arnoldi requires a large initial cost to generate the Arnoldi vectors with subsequently much less expensive dynamics for the time integration. The break‐even point is problem‐dependent at a number of time steps which may be for some problems up to one order of magnitude larger than the number of Arnoldi vectors. A serious limitation of Arnoldi is the requirement of linearity and time independence of the flow field. It is concluded that Arnoldi can be cheaper than Crank–Nicolson in very few instances, i.e. when the solution is needed for a large number of time values, say several hundreds or even 1000, depending on the problem. Copyright © 2001 John Wiley & Sons, Ltd.     

Numeric simulation of pollutant dispersion by a control‐volume based on finite element method

The dispersion of pollutants in the environment is an issue of great interest as it directly affects air quality, mainly in large cities. Experimental and numerical tools have been used to predict the behavior of pollutant species dispersion in the atmosphere. A software has been developed based on the control‐volume based on the finite element method in order to obtain two‐dimensional simulations of Navier–Stokes equations and heat or mass transportation in regions with obstacles, varying position of the pollutant source. Numeric results of some applications were obtained and, whenever possible, compared with literature results showing satisfactory accordance. Copyright © 2010 John Wiley & Sons, Ltd.     

An operator splitting algorithm for the three-dimensional advection–diffusion equation

Operator splitting algorithms are frequently used for solving the advection–diffusion equation, especially to deal with advection dominated transport problems. In this paper an operator splitting algorithm for the three-dimensional advection–diffusion equation is presented. The algorithm represents a second-order-accurate adaptation of the Holly and Preissmann scheme for three-dimensional problems. The governing equation is split into an advection equation and a diffusion equation, and they are solved by a backward method of characteristics and a finite element method, respectively. The Hermite interpolation function is used for interpolation of concentration in the advection step. The spatial gradients of concentration in the Hermite interpolation are obtained by solving equations for concentration gradients in the advection step. To make the composite algorithm efficient, only three equations for first-order concentration derivatives are solved in the diffusion step of computation. The higher-order spatial concentration gradients, necessary to advance the solution in a computational cycle, are obtained by numerical differentiations based on the available information. The simulation characteristics and accuracy of the proposed algorithm are demonstrated by several advection dominated transport problems. © 1998 John Wiley & Sons, Ltd.     

A multi‐dimensional monotonic finite element model for solving the convection–diffusion‐reaction equation

In this paper, we develop a finite element model for solving the convection–diffusion‐reaction equation in two dimensions with an aim to enhance the scheme stability without compromising consistency. Reducing errors of false diffusion type is achieved by adding an artificial term to get rid of three leading mixed derivative terms in the Petrov–Galerkin formulation. The finite element model of the Petrov–Galerkin type, while maintaining convective stability, is modified to suppress oscillations about the sharp layer by employing the M‐matrix theory. To validate this monotonic model, we consider test problems which are amenable to analytic solutions. Good agreement is obtained with both one‐ and two‐dimensional problems, thus validating the method. Other problems suitable for benchmarking the proposed model are also investigated. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical investigations of stabilized finite element computations for acoustics

Least-squares stabilization stands out among the numerous approaches that have been proposed for relaxing resolution requirements of Galerkin computations for acoustics, by combining substantial improvement in performance with extremely simple implementation. The Galerkin/least-squares and Galerkin-gradient/least-squares methods are quite similar for structured meshes of linear finite elements. A series of numerical tests compares the two methods for several configurations with different kinds of boundary conditions employing structured and unstructured meshes. Various definitions of the resolution-dependent stability parameters are considered, along with different definitions of the mesh size upon which they depend.