Non-symmetric CG-like schemes and the finite element solution of the advection–dispersion equation

Seven leading iterative methods for non-symmetric linear systems (GMRES, BCG, QMR, CGS, Bi-CGSTAB, TFQMR and CGNR) are compared in the specific context of solving the advection–dispersion equation by a classic approach: The space derivatives are approximated by linear finite elements while an implicit scheme is used to integrate the time derivatives. Convergence formulas that predict the behaviour of the iterative methods as a function of the discretization parameters are developed and validated by experiments. It is shown that all methods converge nicely when the coefficent matrix of the linear system is close to normal and the finite element approximation of the advection–dispersion equation yields accurate results.     

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.     

An integrated modelling system for coastal area dynamics

Two-dimensional initial-boundary value problems are considered for the shallow water equations and the equation of advection and dispersion of pollutants. The problems are solved in curvilinear boundary fitted co-ordinates. The transformed equations are integrated on a regular grid by the semi-implicit and implicit finite difference methods. Based on the numerical method, the integrated modelling system Cardinal for coastal area dynamics and pollution processes is developed for application on personal computers. Examples of computations are given.     

Discontinuous boundary implementation for the shallow water equations

Quasi‐bubble finite element approximations to the shallow water equations are investigated focusing on implementations of the surface elevation boundary condition. We first demonstrate by numerical results that the conventional implementation of the boundary condition degrades the accuracy of the velocity solution. It is also shown that the degraded velocity leads to a critical instability if the advection term is present in the momentum equation. Then we propose an alternative implementation for the boundary condition. We refer to this alternative implementation as a discontinuous boundary (DB) implementation because it introduces at each boundary node two independent mass–flux values that result in a discontinuity at the boundary. Numerical results show that the proposed DB implementation is consistent, stabilizes the quasi‐bubble scheme, and leads to second‐order accuracy at the surface elevation specified boundary. Copyright © 2004 John Wiley & Sons, Ltd.     

Analytical Solution for Multi-Species Contaminant Transport in Finite Media with Time-Varying Boundary Conditions

Most analytical solutions available for the equations governing the advective–dispersive transport of multiple solutes undergoing sequential first-order decay reactions have been developed for infinite or semi-infinite spatial domains and steady-state boundary conditions. In this study, we present an analytical solution for a finite domain and a time-varying boundary condition. The solution was found using the Classic Integral Transform Technique (CITT) in combination with a filter function having separable space and time dependencies, implementation of the superposition principle, and using an algebraic transformation that changes the advection–dispersion equation for each species into a diffusion equation. The analytical solution was evaluated using a test case from the literature involving a four radionuclide decay chain. Results show that convergence is slower for advection-dominated transport problems. In all cases, the converged results were identical to those obtained with the previous solution for a semi-infinite domain, except near the exit boundary where differences were expected. Among other applications, the new solution should be useful for benchmarking numerical solutions because of the adoption of a finite spatial domain.     

The concept of minimized integrated exponential error for low dispersion and low dissipation schemes

We devise two novel techniques to optimize parameters which regulate dispersion and dissipation effects in numerical methods using the notion that dissipation neutralizes dispersion. These techniques are baptized as the minimized integrated error for low dispersion and low dissipation (MIELDLD) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD) . These two techniques of optimization have an advantage over the concept of minimized integrated square difference error (MISDE) , especially in the case when more than one optimal cfl is obtained, out of which only one of these values satisfy the shift condition. For instance, when MISDE is applied to the 1‐D Fromm's scheme, we have obtained two optimal cfl numbers: 0.28 and 1.0. However, it is known that Fromm's scheme satisfies shift condition only at r=1.0. Using MIELDLD and MIEELDLD , the optimal cfl of Fromm's scheme is computed as 1.0. We show that like the MISDE concept, both the techniques MIELDLD and MIEELDLD are effective to control dissipation and dispersion. The condition ν²>4µ is satisfied for all these three techniques of optimization, where ν and µ are parameters present in the Korteweg‐de‐Vries‐Burgers equation. The optimal cfl number for some numerical schemes namely Lax–Wendroff, Beam–Warming, Crowley and Upwind Leap‐Frog when discretized by the 1‐D linear advection equation is computed. The optimal cfl number obtained is in agreement with the shift condition. Some numerical experiments in 1‐D have been performed which consist of discontinuities and shocks. The dissipation and dispersion errors at some different cfl numbers for these experiments are quantified. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimised composite numerical schemes in 2‐D for hyperbolic conservation laws

One of the techniques available for optimising parameters that regulate dispersion and dissipation effects in finite difference schemes is the concept of minimised integrated exponential error for low dispersion and low dissipation. In this paper, we work essentially with the two‐dimensional (2D) Corrected Lax–Friedrichs and Lax–Friedrichs schemes applied to the 2D scalar advection equation. We examine the shock‐capturing properties of these two numerical schemes, and observe that these methods are quite effective from the point of being able to control computational noise and having a large range of stability. To improve the shock‐capturing efficiency of these two methods, we derive composite methods using the idea of predictor/corrector or a linear combination of the two schemes. The optimal cfl number for some of these composite schemes are computed. Some numerical experiments are carried out in two dimensions such as cylindrical explosion, shock‐focusing, dam‐break and Riemann gas dynamics tests. The modified equations of some of the composite schemes when applied to the 2D scalar advection equation are obtained. We also perform some convergence tests to obtain the order of accuracy and show that better results in terms of shock‐capturing property are obtained when the optimal cfl obtained using minimised integrated exponential error for low dispersion and low dissipation is used. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of One-Dimensional Advection–Diffusion Model with Variable Coefficients Describing Solute Transport in a Porous medium

This work presents the analytical solution and temporal moments of one-dimensional advection–diffusion model with variable coefficients. Two case studies along with the two different sets of boundary conditions are considered at the inlet and outlet of the domain. In the first case, a time-dependent solute dispersion in the homogeneous domain along uniform flow is taken into account, whereas in the second case, due to inhomogeneity of domain, velocity is taken spatially dependent and the dispersion is assumed proportional to the square of the velocity. The Laplace transform is used to obtain the analytical solutions. The analytical temporal moments are derived from the Laplace domain solutions. To verify the correctness of the analytical solutions, a high-resolution second-order finite volume scheme is applied. Different case studies are considered and discussed. Both analytical and numerical results are in good agreement with each other.     

High‐order boundary conditions for linearized shallow water equations with stratification,dispersion and advection

The two‐dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ??, and a high‐order open boundary condition (OBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd.     

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

This paper presents a detailed multi‐methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi‐discretizations of the scalar advection–diffusion equation. The errors are reported in terms of non‐dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid‐induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew‐symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov–Galerkin and its control‐volume finite element analogue, the streamline upwind control‐volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi‐discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super‐convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second‐order behaviour. In Part II of this paper, we consider two‐dimensional semi‐discretizations of the advection–diffusion equation and also assess the affects of grid‐induced anisotropy observed in the non‐dimensional phase speed, and the discrete and artificial diffusivities. Although 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 analysis framework. Published in 2004 by John Wiley & Sons, Ltd.     

Analysis of the local truncation error in the pressure‐free projection method for incompressible flows: a new accurate expression of the intermediate boundary conditions

The numerical integration of the Navier–Stokes equations for incompressible flows demands efficient and accurate solution algorithms for pressure–velocity splitting. Such decoupling was traditionally performed by adopting the Fractional Time‐Step Method that is based on a formal separation between convective–diffusive momentum terms from the pressure gradient term. This idea is strictly related to the fundamental theorem on the Helmholtz–Hodge orthogonal decomposition of a vector field in a finite domain, from which the name projection methods originates. The aim of this paper is to provide an original evaluation of the local truncation error (LTE) for analysing the actual accuracy achieved by solving the de‐coupled system. The LTE sources are formally subdivided in two categories: errors intrinsically due to the splitting of the original system and errors due to the assignment of the boundary conditions. The main goal of the present paper consists in both providing the LTE analysis and proposing a remedy for the inaccuracy of some types of intermediate boundary conditions associated with the prediction equation. Such evaluations will be directly performed in the physical space for both the time continuous formulation and the finite volume discretization along with the discrete Adams–Bashforth/Crank–Nicolson time integration. A new proposal for a boundary condition expression, congruent with the discrete prediction equation is herein derived, fulfilling the goal of accomplishing the closure of the problem with fully second order accuracy. In our knowledge, this procedure is new in the literature and can be easily implemented for confined flows. The LTE is clearly highlighted and many computations demonstrate that our proposal is efficient and accurate and the goal of adopting the pressure‐free method in a finite domain with fully second order accuracy is reached. Copyright © 2003 John Wiley & Sons, Ltd.     

The influence of normal flow boundary conditions on spurious modes in finite element solutions to the shallow water equations

Finite element solutions of the primitive equation (PE) form of the shallow water equations are notorious for the severe spurious 2Δx modes which appear. Wave equation (WE) solutions do not exhibit these numerical modes. In this paper we show that the severe spurious modes in PE solutions are strongly influenced by essential normal flow boundary conditions in the coupled continuity-momentum system of equations. This is demonstrated through numerical examples that avoid the use of essential normal flow boundary conditions either by specifying elevation values over the entire boundary or by implementing natural flow boundary conditions in the weak weighted residual form of the continuity equation. Results from a series of convergence tests show that PE solutions are of nearly the same quality as WE solutions when spurious modes are suppressed by alternative specification of the boundary conditions. Network intercomparisons indicate that varying nodal support does not excite spurious modes in a solution, although it does enhance the spurious modes introduced when an essential normal flow boundary condition is used. Dispersion analysis of discrete equations for interior and boundary nodes offers an explanation of the observed solution behaviour. For certain PE algorithms a mixed situation can arise where the boundary nodes exhibit a monotonic (noise-free) dispersion relationship and the interior nodes exhibit a folded (noisy) dispersion relationship. We have found that the mixed situation occurs when all boundary nodes are specified elevation nodes (which are enforced as essential conditions in the continuity equation) or when specified flow boundary nodes are treated as natural boundary conditions in the continuity equation. In either case the effect is to generate a solution that is essentially free of noise. Apparently, the monotonic dispersion behaviour at the boundaries suppresses the otherwise noisy behaviour caused by the folded dispersion relation on the interior.     

An inverse problem formulation of the immersed-boundary method

We formulate the immersed-boundary method (IBM) as an inverse problem. A control variable is introduced on the boundary of a larger domain that encompasses the target domain. The optimal control is the one that minimizes the mismatch between the state and the desired boundary value along the immersed target-domain boundary. We begin by investigating a naïve problem formulation that we show is ill-posed: in the case of the Laplace equation, we prove that the solution is unique, but it fails to depend continuously on the data; for the linear advection equation, even solution uniqueness fails to hold. These issues are addressed by two complimentary strategies. The first strategy is to ensure that the enclosing domain tends to the true domain, as the mesh is refined. The second strategy is to include a specialized parameter-free regularization that is based on penalizing the difference between the control and the state on the boundary. The proposed inverse IBM is applied to the diffusion, advection, and advection-diffusion equations using a high-order discontinuous Galerkin discretization. The numerical experiments demonstrate that the regularized scheme achieves optimal rates of convergence and that the reduced Hessian of the optimization problem has a bounded condition number, as the mesh is refined.     

A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions

We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.     

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.     

A diffuse-interface immersed boundary method for simulation of compressible viscous flows with stationary and moving boundaries

In this paper, a diffuse-interface immersed boundary method (IBM) is proposed for simulation of compressible viscous flows with stationary and moving boundaries. In the method, the solution of flow field and the implementation of boundary conditions are decoupled into two steps by applying the fractional step technique, ie, the predictor step and the corrector step. Firstly, in the predictor step, the intermediate flow field is resolved by a recently developed gas kinetic flux solver (GKFS) without consideration of the solid boundary. The GKFS is a finite volume approach that solves the Navier-Stokes equations for the flow variables at cell centers. In GKFS, the inviscid and viscous fluxes are evaluated as a single entity by reconstructing the local solution of continuous Boltzmann equation. Secondly, in the corrector step, the intermediate flow field is corrected by the present diffuse-interface IBM. During this process, the velocity field is firstly corrected by the implicit boundary condition–enforced IBM so that the no-slip boundary condition can be accurately satisfied. After that, the density correction is made by an iterative approach with the help of the continuity equation. Finally, the correction of the temperature field is made in the same way as that of the velocity field. Good agreements between the present simulations and the reference data in literature demonstrate the reliability of the proposed method.     

Scattered noise prediction using acoustic velocity formulations V1A and KV1A

Aeroacoustic scattering prediction generally relies on boundary integral methods which require evaluation of the impermeability condition on the scattering surface. The boundary condition implies zero normal velocity relative to the scattering surface. This condition has been expressed by relating the acoustic velocity to the acoustic pressure gradient, allowing indirect evaluation of the boundary condition by existent acoustic pressure gradient formulations. In the present paper, a direct evaluation of the hardwall boundary condition in scattering problems is demonstrated by time-domain analytic acoustic velocity formulae. Acoustic velocity formulations V1A and KV1A are implemented for acoustic scattering prediction, by hybrid approaches based on the FW–H equation and the Kirchhoff method These formulations can be coupled to any scattering solver, allowing time-domain prediction of the incident acoustic field when broadband noise generation is concerned. Formulation V1A offers mathematical simplicity and computational efficiency, which can be advantageous for realistic scattering applications. Implementation of formula KV1A enables acoustic scattering prediction by existing solvers based on the Kirchhoff method. The validity of the suggested methodology is assessed through the analytical test case of harmonic sound scattered by a rigid sphere. Sound propagation and scattering effects are analyzed by examination of the acoustic velocity field characteristics.     

An adaptive enrichment algorithm for advection‐dominated problems

We are interested in developing a numerical framework well suited for advection–diffusion problems when the advection part is dominant. In that case, given Dirichlet type boundary condition, it is well known that a boundary layer develops. To resolve correctly this layer, standard methods consist in increasing the mesh resolution and possibly increasing the formal accuracy of the numerical method. In this paper, we follow another path: we do not seek to increase the formal accuracy of the scheme but, by a careful choice of finite element, to lower the mesh resolution in the layer. Indeed the finite element representation we choose is locally the sum of a standard one plus an enrichment. This paper proposes such a method and with several numerical examples, we show the potential of this approach. In particular, we show that the method is not very sensitive to the choice of the enrichment and develop an adaptive algorithm to automatically choose the enrichment functions.Copyright © 2012 John Wiley & Sons, Ltd.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

Inhomogenous vertical turbulent convection in the planetary boundary type

The inhomogeneous vertical turbulent convection in the planetary boundary layer has been analysed. The governing equation is the three dimensional advection equation. With the help of a model of the temperature-velocity correlation the advection equation is transformed into a solvable equation. Numerical solution has been obtained for some particular conditions.