Generalized fourier analyses of the advection–diffusion equation—Part II: two‐dimensional domains

Part I of this work presents a detailed multi‐methods comparison of the spatial errors associated with the one‐dimensional finite difference, finite element and finite volume semi‐discretizations of the scalar advection–diffusion equation. In Part II we extend the analysis to two‐dimensional domains and also consider the effects of wave propagation direction and grid aspect ratio on the phase speed, and the discrete and artificial diffusivities. The observed dependence of dispersive and diffusive behaviour on propagation direction makes comparison of methods more difficult relative to the one‐dimensional results. For this reason, integrated (over propagation direction and wave number) error and anisotropy metrics are introduced to facilitate comparison among the various methods. With respect to these metrics, the consistent mass Galerkin and consistent mass control‐volume finite element methods, and their streamline upwind derivatives, exhibit comparable accuracy, and generally out‐perform their lumped mass counterparts and finite‐difference based schemes. While this work can only be considered a first step in a comprehensive multi‐methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common mathematical framework. Published in 2004 by John Wiley & Sons, Ltd.     

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.     

Fourier analysis of several finite difference schemes for the one‐dimensional unsteady convection–diffusion equation

This paper reports a comparative study on the stability limits of nine finite difference schemes to discretize the one‐dimensional unsteady convection–diffusion equation. The tested schemes are: (i) fourth‐order compact; (ii) fifth‐order upwind; (iii) fourth‐order central differences; (iv) third‐order upwind; (v) second‐order central differences; and (vi) first‐order upwind. These schemes were used together with Runge–Kutta temporal discretizations up to order six. The remaining schemes are the (vii) Adams–Bashforth central differences, (viii) the Quickest and (ix) the Leapfrog central differences. In addition, the dispersive and dissipative characteristics of the schemes were compared with the exact solution for the pure advection equation, or simple first or second derivatives, and numerical experiments confirm the Fourier analysis. The results show that fourth‐order Runge–Kutta, together with central schemes, show good conditional stability limits and good dispersive and dissipative spectral resolution. Overall the fourth‐order compact is the recommended scheme. Copyright © 2001 John Wiley & Sons, Ltd.     

Insights on a sign‐preserving numerical method for the advection–diffusion equation

In this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one‐dimensional unsteady advection–diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first‐order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second‐order terms of the truncation error and the other considers additionally the third‐order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

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.     

Managing false diffusion during second‐order upwind simulations of liquid micromixing

Liquid mixing is an important component of many microfluidic concepts and devices, and computational fluid dynamics (CFD) is playing a key role in their development and optimization. Because liquid mass diffusivities can be quite small, CFD simulation of liquid micromixing can over predict the degree of mixing unless numerical (or false) diffusion is properly controlled. Unfortunately, the false diffusion behavior of higher‐order finite volume schemes, which are often used for such simulations, is not well understood, especially on unstructured meshes. To examine and quantify the amount of false diffusion associated with the often recommended and versatile second‐order upwind method, a series of numerical simulations was conducted using a standardized two‐dimensional test problem on both structured and unstructured meshes. This enabled quantification of an ‘effective’ false diffusion coefficient (D_false) for the method as a function of mesh spacing. Based on the results of these simulations, expressions were developed for estimating the spacing required to reduce D_false to some desired (low) level. These expressions, together with additional insights from the standardized test problem and findings from other researchers, were then incorporated into a procedure for managing false diffusion when simulating steady, liquid micromixing. To demonstrate its utility, the procedure was applied to simulate flow and mixing within a representative micromixer geometry using both unstructured (triangular) and structured meshes. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Preserving bounded and conservative solutions of transport in one‐dimensional shallow‐water flow with upwind numerical schemes: Application to fertigation and solute transport in rivers

This work intends to show that conservative upwind schemes based on a separate discretization of the scalar solute transport from the shallow‐water equations are unable to preserve uniform solute profiles in situations of one‐dimensional unsteady subcritical flow. However, the coupled discretization of the system is proved to lead to the correct solution in first‐order approximations. This work is also devoted to show that, when using a coupled discretization, a careful definition of the flux limiter function in second‐order TVD schemes is required in order to preserve uniform solute profiles. The work shows that, in cases of subcritical irregular flow, the coupled discretization is necessary but nevertheless not sufficient to ensure concentration distributions free from oscillations and a method to avoid these oscillations is proposed. Examples of steady and unsteady flows in test cases, river and irrigation are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

The steady Navier–Stokes/energy system with temperature‐dependent viscosity—Part 1: Analysis of the continuous problem

In this first part we propose and analyse a model for the study of two‐dimensional incompressible Navier–Stokes equations with a temperature‐dependent viscosity. The flow is supposed in a mixed convection regime and considers an outflow region, leading to a strongly coupled problem between the Navier–Stokes and energy equations, which will be justified theoretically. The coupling in the continuous problem is treated by an outer temperature fixed point strategy. Existence results for a particular variational formulation follows from this study. Further, a particular uniqueness result for small data is also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

The steady Navier–Stokes/energy system with temperature‐dependent viscosity—Part 2: The discrete problem and numerical experiments

In this second part, we analyse the associated discrete problem arising from a conforming finite element method formulation of the mathematical model presented in the first part. Thus, existence and uniqueness of the discrete solution when using small enough data are stated following the same strategy used in the continuous case, with a Cea's type error estimate established as the main result. Some numerical experiments, steady and unsteady, are performed, which allow us to validate the previous mathematical model and its discrete approximation. Copyright © 2007 John Wiley & Sons, Ltd.     

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two‐dimensional unsteady convection–diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5?μ?1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection–diffusion problem and three flows of varying complexities governed by the two‐dimensional incompressible Navier–Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, efficient and accurate. Copyright © 2002 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.