On preconditioned Krylov subspace methods for discrete convection–diffusion problems

We study nonstationary iterative methods for solving preconditioned systems arising from discretizations of the convection–diffusion equation. The preconditioners arise from Gauss–Seidel methods applied to the original system. It is shown that the performance of the iterative solvers is affected by the relationship of the ordering of the underlying grid and the direction of the fow associated with the differential operator. Specifically, only those orderings that follow the fow give fast iterative solvers. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13 :321–330     

Parallel characteristic finite difference method for convection–diffusion equations

Based on the overlapping domain decomposition, an efficient parallel characteristic finite difference scheme is proposed for solving convection‐diffusion equations numerically. We give the optimal convergence order in error estimate analysis, which shows that we just need to iterate once or twice at each time level to reach the optimal convergence order. Numerical experiments also confirm the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 854–866, 2011     

Fast second-order accurate difference schemes for time distributed-order and Riesz space fractional diffusion equations

The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in $L_2$-norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods.     

An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem

An a posteriori upper bound is derived for the nonstationary convection–diffusion problem using the Crank–Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadratic time reconstruction is used.A space and time adaptive algorithm is developed to ensure the control of the relative error in the L²(H¹) norm. Numerical experiments illustrating the efficiency of this approach are reported; it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step, even in the convection dominated regime and in the presence of boundary layers.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014     

On a convection–diffusion problem with a weak layer

In this article we consider a model linear convection–diffusion problem with a weak layer. We analyze the singular-perturbation nature of the problem and show that no special precautions are required to cope with the weak layer: a standard upwind scheme on a (quasi-)uniform mesh is sufficient. We give a simple analysis for the method. Thus highlighting that not all problems with a small parameter multiplying the highest-order derivative are suitable for studying boundary-layer phenomena.     

A study on a second order finite difference scheme for fractional advection–diffusion equations

Second order finite difference schemes for fractional advection–diffusion equations are considered in this paper. We note that, when studying these schemes, advection terms with coefficients having the same sign as those of diffusion terms need additional estimates. In this paper, by comparing generating functions of the corresponding discretization matrices, we find that sufficiently strong diffusion can dominate the effects of advection. As a result, convergence and stability of schemes are obtained in this situation.     

Critical Fujita exponents for a class of nonlinear convection–diffusion equations

In this paper, we establish the blow‐up theorems of Fujita type for a class of homogeneous Neumann exterior problems of quasilinear convection–diffusion equations. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow‐up case under any nontrivial initial data. Copyright © 2010 John Wiley & Sons, Ltd.     

Superconvergence of finite difference approximations for convection–diffusion problems

An Erratum has been published for this article in Numerical Linear Algebra with Applications 8 (4) 2001, iii–iv. We are concerned with numerical solutions of convection–diffusion equations. The convergence behaviour of numerical solutions is considered by using the finite difference approximation with respect to spatial variables and implicit method with respect to time variable. It is shown that superconvergence occurs near a part of the boundary which has Dirichlet's data. Copyright © 2001 John Wiley & Sons, Ltd.     

Chaos gray-coded genetic algorithm and its application for pollution source identifications in convection–diffusion equation

In order to reduce the computational amount and improve computational precision for nonlinear optimizations and pollution source identification in convection–diffusion equation, a new algorithm, chaos gray-coded genetic algorithm (CGGA) is proposed, in which initial population are generated by chaos mapping, and new chaos mutation and Hooke–Jeeves evolution operation are used. With the shrinking of searching range, CGGA gradually directs to an optimal result with the excellent individuals obtained by gray-coded genetic algorithm. Its convergence is analyzed. It is very efficient in maintaining the population diversity during the evolution process of gray-coded genetic algorithm. This new algorithm overcomes any Hamming-cliff phenomena existing in other encoding genetic algorithm. Its efficiency is verified by application of 20 nonlinear test functions of 1–20 variables compared with standard binary-coded genetic algorithm and improved genetic algorithm. The position and intensity of pollution source are well found by CGGA. Compared with Gray-coded hybrid-accelerated genetic algorithm and pure random search algorithm, CGGA has rapider convergent speed and higher calculation precision.     

A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method

A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.     

Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection–diffusion equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods.     

A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

Nonstandard finite difference schemes for Michaelis–Menten type reaction‐diffusion equations

We compare and investigate the performance of the exact scheme of the Michaelis–Menten (M–M) ordinary differential equation with several new nonstandard finite difference (NSFD) schemes that we construct using Mickens' rules. Furthermore, the exact scheme of the M–M equation is used to design several dynamically consistent NSFD schemes for related reaction‐diffusion equations, advection‐reaction equations, and advection‐reaction‐diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.     

Parallel characteristic finite element method for time‐dependent convection–diffusion problem

Based on the overlapping‐domain decomposition and parallel subspace correction method, a new parallel algorithm is established for solving time‐dependent convection–diffusion problem with characteristic finite element scheme. The algorithm is fully parallel. We analyze the convergence of this algorithm, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration times and sub‐domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to optimal accuracy at each time step. Copyright © 2010 John Wiley & Sons, Ltd.     

A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection–diffusion problem

A numerical method is proposed for solving singularly perturbed one-dimensional parabolic convection–diffusion problems. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and B-spline collocation method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O((Δx)²+Δt). An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Several numerical experiments have been carried out in support of the theoretical results. Comparisons of the numerical solutions are performed with an upwind finite difference scheme on a piecewise uniform mesh and exponentially fitted method on a uniform mesh to demonstrate the efficiency of the method.     

Quasi‐Toeplitz splitting iteration methods for unsteady space‐fractional diffusion equations

We construct a class of quasi‐Toeplitz splitting iteration methods to solve the two‐sided unsteady space‐fractional diffusion equations with variable coefficients. By making full use of the structural characteristics of the coefficient matrix, the method only requires computational costs of O(n log n) with n denoting the number of degrees of freedom. We develop an appropriate circulant matrix to replace the Toeplitz matrix as a preconditioner. We discuss the spectral properties of the quasi‐circulant splitting preconditioned matrix. Numerical comparisons with existing approaches show that the present method is both effective and efficient when being used as matrix splitting preconditioners for Krylov subspace iteration methods.     

An exponential high‐order compact ADI method for 3D unsteady convection–diffusion problems

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013