Discrete Convection-diffusion Equation Research Articles

Formation and arrangement of pits by a corrosive gas.

When corroding or otherwise aggressive particles are incident on a surface, pits can form. For example, under certain circumstances rock surfaces that are exposed to salts can form regular tessellating patterns of pits known as "tafoni." We introduce a simple lattice model in which a gas of corrosive particles, described by a discrete, biased diffusion equation, drifts onto a surface. Each gas particle has a fixed probability of being absorbed and causing damage at each contact. The surface is represented by a lattice of strength numbers which reduce after each absorbtion event, with sites being removed when their strength becomes negative. Regular formations of pits arise spontaneously, with each pit having a characteristic trapezoidal geometry determined by the particle bias, absorbtion probability, and surface strength. The formation of this geometry may be understood in terms of a first order partial differential equation and is a consequence of particle concentration gradients which arise in the pits. By viewing pits as particle funnels, we are able to relate the gradient of pit walls to absorbtion probability and particle bias.

A class of preconditioners based on matrix splitting for nonsymmetric linear systems

Given a general matrix splitting A = M - N where M is nonsingular, a new factorization scheme in terms of factorized and splitting matrices is given using the Sherman–Morrison formula. Theoretical analysis shows that the factorization can give an LDU decomposition of A under some special choices. We propose and implement a class of preconditioners based on this factorization combining with dropping rules. A number of numerical experiments from discrete convection diffusion equation and some practical problems show that the new preconditioner is efficient, and is comparable to existing preconditioners in term of storage requirement and computational cost.

Fourier Analysis of Multigrid for a Model Two-Dimensional Convection-Diffusion Equation

We present a Fourier analysis of multigrid for the two-dimensional discrete convection-diffusion equation. For constant coefficient problems with grid-aligned flow and semi-periodic boundary conditions, we show that the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4×4 blocks. This enables a trivial computation of the norm of the iteration matrix demonstrating rapid convergence in the case of both small and large mesh Peclet numbers, where the streamline-diffusion discretisation is used in the latter case. We also demonstrate that these results are strongly correlated with the properties of the iteration matrix arising from Dirichlet boundary conditions.

Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation

We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {A n (a,p)} n discretizing the elliptic (convection-diffusion) problem with Ω being a plurirectangle of R d with a(x) being a uniformly positive function and p(x) denoting the Reynolds function: here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence {P n (a)} n , P n (a):= D n 1/2(a)A n (1,0) D n 1/2(a) where D n (a) is the suitably scaled main diagonal of A n (a,0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {P n (a)} n turns out to be a superlinear preconditioning sequence for {A n (a,0)} n where A n (a,0) represents a good approximation of Re(A n (a,p)) namely the real part of A n (a,p). The computational interest is due to the fact that the preconditioned HSS method has a convergence behavior depending on the spectral properties of {P n -1(a)Re(A n (a,p))} n ≈ {P n -1(a)A n (a,0)} n : therefore the solution of a linear system with coefficient matrix A n (a,p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {A n (1,0)} n . Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.

Ordering Effects on Relaxation Methods Applied to the Discrete One-Dimensional Convection-Diffusion Equation

The authors present an analysis of relaxation methods for the one-dimensional discrete convection-diffusion equation based on norms of the iteration matrices. In contrast to standard analytic techniques that use spectral radii, these results show how the performance of iterative solvers is affected by directions of flow associated with the underlying operator, and by orderings of the discrete grid points. In particular, for problems of size n, relaxation against the flow incurs a latency of approximately n steps in which convergence is slow, and red-black relaxation incurs a latency of approximately ${n / 2}$ steps. There is no latency associated with relaxation that follows the flow. These results are largely independent of the choice of discretization.

Line Iterative Methods for Cyclically Reduced Discrete Convection-Diffusion Problems

An analytic and empirical study of line iterative methods for solving the discrete convection-diffusion equation is performed. The methodology consists of performing one step of the cyclic reduction method, followed by iteration on the resulting reduced system using line orderings of the reduced grid. Two classes of iterative methods are considered: block stationary methods, such as the block Gauss–Seidel and SOR methods, and preconditioned generalized minimum residual methods with incomplete LU preconditioners. New analysis extends convergence bounds for constant coefficient problems to problems with separable variable coefficients. In addition, analytic results show that iterative methods based on incomplete LU preconditioners have faster convergence rates than block Jacobi relaxation methods. Numerical experiments examine additional properties of the two classes of methods, including the effects of direction of flow, discretization, and grid ordering on performance.