Dimensional Convection Diffusion Equation Research Articles

A Spectral Method for Convection-Diffusion Equations

In the practical problems such as nuclear waste pollution and seawater intrusion etc., many problems are reduced to solving the convection-diffusion equation, so the research of convection-diffusion equation is of great value. In this work, a spectral method is presented for solving one and two dimensional convection-diffusion equation with source term. The finite difference method is also used to solve the convection diffusion equation. The numerical experiments show that the spectral method is more efficient than other methods for solving the convection-diffusion equation.

The mass-preserving domain decomposition scheme for solving three-dimensional convection–diffusion equations

In this paper, by combining the operator splitting and second order modified upwind technique, the mass-preserving domain decomposition method for solving time-dependent three dimensional convection–diffusion equations is analyzed. A three steps (x−direction, y−direction and z−direction) method is used to compute the solutions over each non-overlapping sub-domains at each time interval. The intermediate fluxes on the interfaces of sub-domains are firstly computed by the modified semi-implicit flux schemes. Then, the solutions and fluxes in the interiors of sub-domains are computed by the modified-upwind splitting implicit solution and flux coupled schemes. By rigorous mathematical analysis, we proved that our scheme is stable in discrete L2-norm with the restriction on the mesh step h=γ(Δt)2∕3. We give the error estimates and obtain the optimal convergence. Numerical experiments are presented to illustrate convergence and conservation.

Cubic quasi-interpolation spline collocation method for solving convection–diffusion equations

In this paper, we use a cubic spline collocation method to solve a two dimensional convection–diffusion equation. More precisely, we approximate first and second order partial derivatives by those of cubic spline quasi-interpolants to produce a system of first order ordinary differential equations. The resulting system can be solved using MATLAB’s ode solver. Error estimates of quasi-interpolants which are used are given with full discussion. Furthermore, numerical examples are presented to show the validity of our methods.

The Studying of Random Cauchy Convection Diffusion Models under Mean Square and Mean Fourth Calculus

The random partial differential equations have a wide range of physical, chemical, and biological applications. The finite difference method offers an attractively simple approximations for these equations. In this paper, the finite difference technique is performed in order to find an approximation solutions for the linear one dimensional convection-diffusion equation with random variable coefficient. We study the consistency and stability of the finite difference scheme under mean square sense. A statistical measure such as mean for the numerical approximation, and the exact solution based on different statistical distributions is computed.

一个二维对流扩散方程源项反问题的条件唯一性

本文应用变分伴随方法研究二维对流扩散方程中确定线性连续源项的一个反问题，并基于正问题的伴随问题，建立了一个联系已知数据与未知边界流量的变分恒等式，进而利用对伴随问题的近似控制，在反问题的解满足局部保序性的条件下，证明反问题解的唯一性。 This article deals with an inverse problem of determining a continuous linear source term in the two dimensional convection-diffusion equation by using the variational adjoint method. A variational identity connecting the known data with the unknown is established based on an adjoint problem, and the conditional uniqueness to the inverse source problem is proved by the approximate controllability to the adjoint problem under the condition that the inverse solution can keep orders locally.

Higher order ADI method with completed Richardson extrapolation for solving unsteady convection–diffusion equations

A higher order alternating direction implicit (ADI) method with completed Richardson extrapolation is proposed for solving unsteady two dimensional convection–diffusion equations. The method has sixth-order accuracy in space and fourth-order accuracy in time. Completed Richardson extrapolation is used to improve the accuracy of the solution obtained from the high-order ADI method in spatial and temporal domains simultaneously. Stability analysis is given to show the influence of Richardson extrapolation on the unconditionally stable ADI method. Numerical experiments are conducted to test its high accuracy and to compare it with Karaa–Zhang’s high-order ADI method.

Global existence and decay estimate of solution to one dimensional convection-diffusion equation

We study the global existence of solution to one dimensional convection-diffusion equation. Through constructing a Cauchy sequence in a Banach space, we get the local existence of solution to the equation. Based on the global bounds of the solution, we extend the local one to a global one that decays in Hl space.

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.

Fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3D convection–diffusion equation

An improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection–diffusion equation is presented. The novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation. The new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method (Wang and Zhang, 2010) [16] to obtain high order solutions. Since the proposed method needs a fourth order compact scheme with unequal-meshsize grids, a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection–diffusion equation. Numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure.

Fast Iterative Solver for the 2-D Convection-Diffusion Equations

In this paper, we introduce the preconditioned Explicit Decoupled Group (EDG) for solving the two dimensional Convection-Diffusion equation with initial and Dirichlet boundary conditions. The purpose of this paper is to accelerate the convergence rate of the Explicit Decoupled Group (EDG) method by using suitable preconditioned iterative scheme for solving the Convection-Diffusion. The robustness of these new formulations over the existing EDG scheme demonstrated through numerical experiments.

Flow-controlled densification and anomalous dispersion of E. coli through a constriction

Dispersion and migration of bacteria under flow in tortuous and confined structures such as porous or fractured materials is related to a large spectrum of practical interest, but is still poorly understood. Here, we address the question of transport and dispersion of an E. coli suspension flowing through a micro-fluidic channel with a funnel-like constriction in its center. We show a counter-intuitive symmetry breaking of the bacterial concentration, which increases significantly past the funnel. This concentration enhancement persists over large distances from the funnel and disappears at large flow rate values. We map our results onto a one dimensional convection–diffusion equation predicting quantitatively the experimental results, without free parameters, when a conservative non-local source term is introduced. This last term, measured experimentally, represents a long range memory effect due to the unbalance of wall adsorption and desorption processes past the constriction. Our model experiment points out the generic importance of considering such constriction effects in the description of transport properties of active matter in porous media. It also opens the possibility to control the concentration of bacterial suspensions in micro-fluidic channels by simply tuning the flow intensity or direction.

On the using cumulant expansion method and van Kampen’s lemma for stochastic differential equations with forcing

Second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Second-order exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Second-order exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not second-order exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not second-order exact. The significance of the new terms found due to the second-order exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated.

Using Distributed Computers to Deterministically Approximate Higher Dimensional Convection Diffusion Equations

A new approach to solving D > 3 spatial dimensional convection-diffusion equation on clusters of workstations is derived by exploiting the stability and scalability of the combination of a generalized D dimensional high-order compact (HOC) implicit finite difference scheme and parallelized GMRES(m). We then consider its application to multifactor Option pricing using the Black–Scholes equation and further show that an isotropic fourth order compact difference scheme is numerically stable and determine conditions under which its coefficient matrix is positive definite. The performance of GMRES(m) on distributed computers is limited by the inter-processor communication required by the matrix-vector multiplication. It is shown that the compact scheme requires approximately half the number of communications as a non-compact difference scheme of the same order of truncation error. As the dimensionality is increased, the ratio of computation that can be overlapped with communication also increases. CPU times and parallel efficiency graphs for single time step approximation of up to a 7D HOC scheme on 16 processors confirm the numerical stability constraint and demonstrate improved parallel scalability over non-compact difference schemes.

Adaptive computations of a posteriori finite element output bounds: a comparison of the “hybrid-flux” approach and the “flux-free” approach

Two different domain decomposition approaches for the a posteriori finite element output bound method with adaptive subdomain refinement are investigated for the three dimensional convection–diffusion equation. The bound method provides relevant, inexpensive, rigorous lower and upper bounds for output linear functionals of partial differential equation (PDE) associated with a fine mesh at a fraction of the computational cost of the fine mesh solution. To achieve a desired bound-gap (i.e., difference between upper and lower output bounds) at the lowest cost, an adaptive subdomain refinement technique is used to refine subdomains only where needed. The bound method based on a finite element domain decomposition technique requires two global calculations and a multitude of subdomain computations. There are two main approaches to decompose a global domain into local subdomains for the subdomain based output bound method: a hybrid-flux (elemental Neumann subdomain) approach and a flux-free (nodal Neumann subdomain) approach. In this paper, the two different domain decomposition approaches are briefly described and compared. Linear and quadratic finite element discretizations for the hybrid-flux and flux-free approaches are developed and validated for a convergence test and an adaptive subdomain refinement example. The comparison of the hybrid-flux and flux-free approaches is presented in terms of numerical accuracy and computational cost.

Truncation error and oscillation property of the combined compact difference scheme

In this work, we study a sixth order combined compact difference scheme for a one dimensional convection diffusion equation. We derive the truncation error representation and analyze its oscillation property. Numerical experiments are performed to illustrate and support our analysis.

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 ).

Wavelet transform adapted to an approximate Kalman filter system

Wavelets are constructed as a class of orthonormal functions with compact support and multi-resolution. Wavelet transforms can effectively compress data and extract key features in frequencies and locations. Wavelets can also be used as a base of spectral representation in solving PDE's. This work demonstrates an effort to apply wavelet methods to a simple data assimilation system. The multi-resolution wavelet transform is applied to one dimensional convection diffusion equation, and a propagator in wavelet space is constructed. The wavelet transform is also applied to the error covariance matrix. The matrix is then compressed and then truncated, so as to reduce the cost of covariance propagation. The results show that the characteristic and localized features in the flow and error covariance are resolved and captured in this approximated Kalman filter system.

Preconditioned iterative methods and finite difference schemes for convection–diffusion

We conduct experimental study on numerical solution of the two dimensional convection–diffusion equation discretized by three finite difference schemes: the traditional central difference scheme, the standard upwind scheme and the fourth-order compact scheme. We study the computed accuracy achievable by each scheme, the algebraic properties of the coefficient matrices arising from different schemes and the performance of the Gauss–Seidel iterative method, the preconditioned GMRES iterative method, and the multigrid method, for solving linear systems arising from these schemes.

A note on Hermitian splitting induced relaxation methods for convection-diffusion equations

The solution of the linear system Ax = b by iterative methods requires a splitting of the coefficient matrix in the form A = M − N where M is usually chosen to be a diagonal or a triangular matrix. In this article we study relaxation methods induced by the Hermitian and skew-Hermitian splittings for the solution of the linear system arising from a compact fourth order approximation to the one dimensional convection-diffusion equation and compare the convergence rates of these relaxation methods to that of the widely used successive overrelaxation (SOR) method. Optimal convergence parameters are derived for each method and numerical experiments are given to supplement the theoretical estimates. For certain values of the diffusion parameter, a relaxation method based on the Hermitian splitting converges faster than SOR. For two-dimensional problems a block form of the iterative algorithm is presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 581–591, 1998

An operator splitting method for nonlinear convection-diffusion equations

We present a semi-discrete method for constructing approximate solutions to the initial value problem for the \(m\)-dimensional convection-diffusion equation \(u_{t}+\nabla\cdot \vec F(u) =\varepsilon\Delta u\). The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case \(m>1\), dimensional splitting is used to reduce the \(m\)-dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions.