Linear Convection-diffusion Equation Research Articles

Fundamental Systems of Numerical Schemes for Linear Convection-Diffusion Equations and their Relationship to Accuracy

A new approach towards the assessment and derivation of numerical methods for convection dominated problems is presented, based on the comparison of the fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci.

Studies on Nodal Integral Methods for the Convection-Diffusion Equation

The computational efficiencies of two nodal integral methods for the numerical solution of linear convection-diffusion equations are studied. Although the first, which leads to a second-order spatial truncation error, has been reported earlier, it is reviewed in order to lead logically to the development here of the second, which has a third-order error. This third-order nodal integral method is developed by introducing an upwind approximation for the linear terms in the “pseudo-sources” that appear in the transverse-averaged equations introduced in the formulation of nodal integral methods. This upwind approximation obviates the need to develop and solve additional equations for the transverse-averaged first moments of the unknown, as would have to be done in a more straightforwardly developed higher-order nodal integral method. The computational efficiencies of the second-order nodal method and the third-order nodal method—of which there are two versions: one, a full third-order method and the other, which uses simpler second-order equations near the boundaries—are compared with those of both a very traditional method and a recently developed state-of-the-art method. Based on the comparisons reported here for a challenging recirculating flow benchmark problem it appears that, among the five methods studied, the second-order nodal integral method has the highest computational efficiency (the lowest CPU computing times for the same accuracy requirements) in the practical 1% error regime.

Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H 1 finite volume space. We actually prove the convergence of the scheme in a discrete H 1 norm, with an error estimate of order O(h) (on meshes of size h ).

Pointwise Estimates on the Green's Function for a Scalar Linear Convection–Diffusion Equation

Pointwise estimates are found on Green's functions for the scalar linear convection–diffusion equations that arise when a scalar conservation law with non-constant diffusion is linearized about a viscous shock profile of arbitrary strength. The estimates take the form of Gaussian kernels centered around paths determined by the (typically different) asymptotic states of the convection function. The analysis extends the spectral transform method to the non-constant coefficient case.

Generalized trapezoidal formulas for convection-diffusion equations

We describe new time-integration schemes for the linear convection-diffusion equation and for the (viscous) Burgers' equation, employing the generalized trapezoidal formulas of Chawla et al [2]. The obtained generalized trapezoidal formula finite-difference schemes (GTF(α)-FDS) are second order in both time and space and unconditionally stable. The better known existing schemes employ the Euler, the backward Euler or the classical arithmetic-mean trapezoidal formula (AM-TF) for integration in time. The performance of our present GTF(α)-FDS is compared with the AM-TF scheme by considering three test problems wherein the significance of the role played by the parameter α becomes evident in providing both stability and accuracy of the computed solution in the presence of diffusivity.

Multidomain implicit numerical scheme

A multidomain method for the solution of elliptic CFD problems with an ADI scheme is described. Two methods of treatment of internal boundary conditions for ADI functions are discussed, namely an explicit and a semi‒implicit method. Stability conditions for the proposed methods are derived theoretically. The semi‒implicit scheme is more stable than the explicit scheme, leading to improved numerical efficiency for multidomain computations. Numerical computations for a linear convection-diffusion equation, for buoyancy‒driven recirculating flow in a square cavity and for turbulent flow in a square duct confirmed the theoretical results. Computer runs of the multidomain code in a distributed memory multiprocessor system were successful and efficient and produced reliable results. © 1997 John Wiley & Sons, Ltd.

An Optimal Control Analysis of Water Pollution Problem

This paper presents an optimal control analysis applied to water pollution control using a combination of the finite element method and optimal control theory. A numerical model of water pollution can be expressed in terms of the linear two-dimensional shallow water and convection diffusion equations. These equations can be solved by the explicit Euler scheme. The method presented by Sakawa-Shindo is effectively used to implement the control theory. Considering the closed water region such as a lake or a coastal sea, into which some river is flowing, it is shown that the pollutant level in this region can be controlled by controlling the inflow velocity of the river. This paper describes the water pollution control problem with the use of inequality constraints.

A New Finite Variable Difference Method with Application to Locally Exact Numerical Scheme

A new finite variable difference method (FVDM) on finite differencing is presented. The essence of this method consists in determining the optimum spatial difference such that the total variance of the solution is a minimum under the condition that characteristic roots of the resulting difference equation are always nonnegative to ensure the numerical stability. The present FVDM is applied to the locally exact numerical scheme (LENS). The optimum spatial difference of the LENS is derived in terms of local mesh Reynolds numbers. By using this optimum spatial difference the numerical accuracy of the LENS for the linear convection–diffusion equations is increased without numerical oscillations for all mesh Reynolds numbers. The present study suggests that an optimum spatial difference from the viewpoint of numerical stability and accuracy exists according to the numerical schemes.

Stability analysis for a family of space one dimensional convection diffusion difference schemes

This paper concerns theL2-stability analysis for a family of finite difference schemes for the initial value problem of space 1 dimensional linear constant coefficient convection diffusion equation. The space difference operators of the schemes have the order of accuracy greater than or equal to 2. In this paper we say that the scheme satisfies the von Neumann condition if the maximum of the absolute value of the amplification coefficient of the scheme is less than or equal to 1. The problem is to characterize, in terms of the space mesh and the time mesh, the scheme which satisfies the von Neumann condition. To investigate the toughness of the scheme in the computation of the modes in a range of relatively shorter wave length in the relevant discrete problem, the concept of e-stability is introduced, and characterized in this paper. The principal aim of the paper is to give the proof of the characterization.

Optimized Locally Exact Numerical Scheme Based on Finite Variable Difference Method and Characteristic Polynomial Analysis Method. In Case of Convection-Diffusion Equation without Source Terms.

A new method of Finite Variable Difference Method (FVDM) is presented. The feature of this method exists in a procedure to determine the finite spatial difference, in which the total deviation of the numerical solution from the exact solution is minimized, under the condition that roots of the resulting characteristic equation are always non-negative to insure numerical stability. The optimum spatial difference of the LECUSSO scheme for the linear convection-diffusion equation is numerically derived in terms of mesh Reynolds numbers. This optimization highly improves the numerical accuracy of the LECUSSO scheme for linear convection-diffusion equations without numerical oscillations at sufficiently large mesh Reynolds numbers of up to 1,000.

Optimized Locally Exact Numerical Scheme Based on Finite Variable Difference Method and Characteristic Polynomial Analysis Method

A new method of Finite Variable Difference Method (FVDM) is presented. The feature of this method exists in a procedure to determine the finite spatial difference, in which the total deviation of the numerical solution from the exact solution is minimized, under the condition that roots of the resulting characteristic equation are always non-negative to insure numerical stability. The optimum spatial difference of the LECUSSO scheme for the linear convection-diffusion equation is numerically derived in terms of mesh Reynolds numbers. This optimization highly improves the numerical accuracy of the LECUSSO scheme for linear convection-diffusion equations without numerical oscillations at sufficiently large mesh Reynolds numbers of up to 1,000.

CONTROLLED VARIATION SCHEME IN A SEQUENTIAL SOLVER FOR RECIRCULATING FLOWS, PART I: THEORY AND FORMULATION

The formalism of the total variation diminishing (TVD) schemes is utilized to design a convection scheme for incompressible recirculating flows. The scheme has been named the controlled variation scheme (CVS). Even though the CVS does not possess the TVD property for sequential solution algorithms, due to the appearance of source terms, the concept of controlled variation fluxes can be effective in suppressing spurious oscillations that commonly occur in convection-dominated viscous flows, by injecting a nonlinear numerical diffusion, similar to the original TVD schemes, into the central difference scheme. This is demonstrated by using the one-dimensional linear convection-diffusion equation with and without a source term as model problems. The formulation and an efficient implementation of the CVS in a sequential pressure-based solver for incompressible steady-state Navier-Stokes equations is presented in this work. The applications of the CVS for two-dimensional laminar and turbulent flows is presented ...

Optimal error estimation for Petrov-Galerkin methods in two dimensions

We examine the optimality of conforming Petrov-Galerkin approximations for the linear convection-diffusion equation in two dimensions. Our analysis is based on the Riesz representation theorem and it provides an optimal error estimate involving the smallest possible constantC. It also identifies an optimal test space, for any choice of consistent norm, as that whose image under the Riesz representation operator is the trial space. By using the Helmholtz decomposition of the Hilbert space [L 2(Ω)]2, we produce a construction for the constantC in which the Riesz representation operator is not required explicitly. We apply the technique to the analysis of the Galerkin approximation and of an upwind finite element method.

Semidiscrete least squares methods for linear convection-diffusion problem

In this paper, some approximate methods for solving linear convection-diffusion problems are presented. The methods consist in discretizing with respect to time and solving the resulting convection dominated elliptic problem for fixed time by least squares finite element methods. An analysis of least squares approximations is given, including optimal order estimates for piecewise polynomial approximation spaces. The model problem considered is the time-dependent convection dominated linear convection-diffusion equation. Numerical results for the Burgers' equation will also be presented.

A particle method to solve the Navier-Stokes system

We extend to the case of the two-dimensional Navier-Stokes equations, a particle method introduced in a previous paper to solve linear convection-diffusion equations. The method is based on a viscous splitting of the operator. The particles move under the effect of the velocity field but are not affected by the diffusion which is taken into account by the weights. We prove the stability and the convergence of the method.

Conjectures on certain boundary-layer equations and natural coordinates

The problem of exclusion of steady downward unsaturated seepage from underground cavities is reducible to a linear convection—diffusion equation with a no normal-flux condition at the cavity surface. Various exact solutions indicate that a roof boundary-layer analysis centred on the upstream stagnation point, and neglecting peripheral variation, gives to remarkable accuracy the quantity θ max , the crucial dimensionless potential determining whether or not water enters the cavity. The great accuracy of this analysis is attributed to the use of curvilinear coordinates natural to the cavity configuration. Global information (such as up to three separate characteristic lengthscales) is injected into the localized boundary-layer formulation via the metric coefficient of the natural coordinates. These are essential to the boundary-layer analysis. Cartesian coordinates, on the other hand, invariably suggest that no boundary layer exists! Definition of the natural coordinates is discussed and means of constructing them about arbitrary cavities are developed. Results for smooth cavities support the conjecture that roof geometry near the upstream stagnation point largely determines θ max , with downstream details unimportant. Comparison of solutions for flat-roofed rectangular and cylindrical cavities with those for strips and discs indicate, however, that the conjecture applies only in weak form to cavities of polygonal cross-section.

Solution of the transport equations using a moving coordinate system

A convection-diffusion equation arises from the conservation equations in miscible and immiscible flooding, thermal recovery, and water movement through desiccated soil. When the convection term dominates the diffusion term, the equations are very difficult to solve numerically. Owing to the hyperbolic character assumed for dominating convection, inaccurate, oscillating solutions result. A new solution technique minimizes the oscillations. The differential equation is transformed into a moving coordinate system which eliminates the convection term but makes the boundary location change in time. We illustrate the new method on two one-dimensional problems: the linear convection-diffusion equation and a non-linear diffusion type equation governing water movement through desiccated soil. Transforming the linear convection diffusion equation into a moving coordinate system gives a diffusion equation with time dependent boundary conditions. We apply orthogonal collocation on finite elements with a Crank-Nicholson time discretization. Comparisons are made to schemes using fixed coordinate systems. The equation describing movement of water in dry soil is a highly non-linear diffusion-type equation with coefficients varying over six orders of magnitude. We solve the equation in a coordinate system moving with a time-dependent velocity, which is determined by the location of the largest gradient of the solution. The finite difference technique with a variable grid size is applied, and a modified Crank-Nicholson technique is used for the temporal discretization. Comparisons are made to an exact solution obtained by similarity transformation, and with an ordinary finite difference scheme on a fixed coordinate system.