A Method for Improving Line Successive Overrelaxation in Anisotropic Problems-A Theoretical Analysis
- J.W. Watts (Scientific Software Corp.)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- April 1973
- Document Type
- Journal Paper
- 105 - 118
- 1973. Society of Petroleum Engineers
- 5.5 Reservoir Simulation
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A recent paper describes a method of improving the convergence rate of line successive overrelaxation (LSOR) in anisotropic problems. That paper presents computed results showing the method to cause dramatic improvements, but includes no theoretical analysis of the method.
This paper offers such an analysis, explaining mathematically how the method works and why it has the effect it does. In doing so, it clears up, at least partially, two important uncertainties left by the previous paper - when to use the method and how to arrive at an optimal extrapolation parameter.
Briefly, the analysis shows that the method removes the components of certain eigenvectors from the solution vector. As shown by the analysis, sometimes this removal causes a tremendous improvement in the rate of convergence and at other times it has no effect. This indicates, in most cases, whether or not to use the method.
Furthermore, the analysis shows that in certain simple problems the method is really a special case of LSOR. In these cases, the optimal extrapolation parameter can be calculated exactly. In other parameter can be calculated exactly. In other cases, it can be estimated using the same techniques as used with LSOR.
One step in the calculations involved in reservoir simulation is the solution of a large set of simultaneous equations. Often this solution is the most time-consuming part of the simulation.
There are several methods used for solving these equations. One of the more common of these is line successive overrelaxation or LSOR. A recent paper describes a way to accelerate the convergence rate of LSOR. That paper shows the modified method, called LSORC, to be very good in an anisotropic case, but no better than LSOR in an isotropic case.
Briefly, LSORC consists of LSOR, with the lines oriented along columns, combined with a correction made by columns. This correction is made in such a manner that the sum of residuals in each column of mesh points is forced to zero.
In this paper we analyze LSORC as applied to a homogeneous but not necessarily isotropic Neumann problem. We show that the correction made using problem. We show that the correction made using LSORC removes the components of certain eigenvectors of the LSOR iteration matrix from the solution vector. Removing the components of these eigenvecton eliminates the effects of their associated eigenvalues on the rate of convergence. In isotropic cases these eigenvalues have little or no effect on the convergence rate, so no improvement is obtained. In anisotropic cases they determine the rate, and removing their effects causes large improvements in convergence rate.
This analysis clears up two points not covered in the original paper. First, it helps to determine when LSORC should and should not be used. Second, it shows that LSORC is really a special form of LSOR. Therefore, methods used to estimate an optimal extrapolation parameter for LSOR apply to LSORC as well.
We want to consider steady-state single-phase Darcy flow in a rectangle. The length of this rectangle in the x-direction is Lx; in the y-direction, Ly. The boundaries of this rectangle are sealed, so we are considering a Neumann problem. The equation describing this problem is
where qn is a point source or sink term.
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