Solving Approximations To The Convection Diffusion Equation

Publisher Website
Google Scholar
Abstract
Several numerical algorithm for solving a discretized version of the convection-diffusion equation are examined. The algorithms considered areiterative methods derived from the conjugate gradient method, with preconditioning by point - SSOR and incomplete LU factorization. The results ofnumerical experiments with the algorithms, for various values of the coefficient of the convection term and various mesh sizes, are reported. We consider algorithms for solving a finite difference discretization of the convection-diffusion equation(P)â?'(uxx+u yy)+Bux=f on the unit square with homogeneous Dirichlet boundary conditions. For the discretization of (P), we use the modified upwind scheme described by Axelsson and Gustafsson [1].[2/(1+(1/2)Bh)+2+Bh] u(i,j)â?'[1/(1+(1/2)Bh)]u(i+1,j)â?'u(i,j+1)â?'[1/(1+(1/2)Bh)â?'Bh] u(iâ?'1,j)â?'u(i,jâ?'1)=f(i,j). which provides order of accuracy 0(h**2), where h is the mesh size. If the coefficient B of the convection term is nonzero, then the above system of linear equations is nonsymmetric. The conjugate gradient algorithm for solving systems of linear equationsAx=b requires that the matrix A be symmetric and positive definite. This paper reports the results of some numerical experiments run on a collection of algorithms modeled after the conjugate gradient algorithm for which it issufficient that the symmetric part M of A,M=(1/2) (A+AT), be positive definite, Three classes of algorithms are considered:a collection of algorithms modeled after the conjugate residual algorithm(described below);the generalized conjugate gradient algorithm developed by Concus and Golub[3] and Widlund [9] andthe conjugate gradient algorithm applied to the normal equationsATAx=ATb. The conjugate residual algorithm for the symmetric problem consists of the following iteration:Choose x(0).Compute r(0)=bâ?'Ax(0)Set p(0)=r(0). For i=0,1,2, . . ., computea(i)=(r(i),Ar(i)/(Ap(i),ap(i)),x(i+1)=x(i)+a(i)p(i),r(i+1)=r(i)â?'a(i)Ap(i),b(i)=(r(i+1),Ar(i+1))/(r(i),Ar(i))p(i+1)=r(i+1)+b(i)p(i) Ignoring roundoff error, this algorithm produces the solution to the system of equations in at most N iterations, where N is the size of the system (see Chandra, [2]). Each direction vector p(i) is ATA-orthogonal to the previous directions computed. The choice of a(i) minimizes the L 2-norm of the residual r(i+1) considered as a function of a(i). x(i+1) is also the point in the affine spacex(0)+ which minimizes the L2-norm of the residual.

This publication has 0 references indexed in Scilit: