Difference corrections and Extrapolation

The difference scheme given above has been shown to yield an approximation to the solution of the BVP to within an error that is . We shall briefly examine two ways, in which, with additional calculations, the difference scheme can be made to yield accuracy. These error reduction procedures are Richardson's deferred approach to the limit or as we prefer to call it, extrapolation to zero mesh width, and the method of difference correction.

The theoretical basis for both methods is the same, namely that some function , independent of the mesh spacing , such that the error has the form

(10.8)

Suppose, we compute , an approximation to ; then clearly

is an approximation to on the mesh. This is essentially the difference correction method and there may be various ways in which the can be determined.

For the extrapolation, we solve the difference system twice, with the net spacing and . Let the respective solutions of these difference problems be denoted by and . For any point common to both meshes, say , we have from (10.8)

.

Thus an approximation to on the net with spacing is given by

.

A derivation of (10.8) is contained in the proof of the following: