1549301742-The_Theory_of_Difference_Schemes__Samarskii

560 Economical Difference Schen1es for Multidirnensional Problen1s

This 1neans that the second term 0( r^2 ) in the available exppression for p

will be excluded from further consideration. The proof of this statement

is omitted here. As a matter of fact, the stability of the scheme at hand

with respect to the boundary conditions is revealed through such a stability

analysis.

It is worth mentioning here several things for later use. Scheme (33)

with the boundary conditions ( 45) is in common usage for "step-shaped"

regions G, whose sides are parallel to the coordinate axes. In the case of an

arbitrary domain this scheme is of accuracy O(lhl^2 + r^2 ../h,). Scheme (9)-

( 10) cannot be forn1ally generalized for the three-dimensional case, smce

the instability is revealed in the resulting scheme.

6. A higher-accuracy sche1ne. By n1inor changes we are led to a higher-
   accuracy sche1ne such as

( 46)

J:Ewh, n=0,1,2, ... , y(:c,0)=u 0 (x), xEwh,

( 47)

where

(} CY --^1 h

2

----CY

2 12 T'

It is plain to show that the scheme in view is stable and generates an
approximation ofO(r^2 +lhl^4 ). With this ai1n, it seems worthwhile to design
a factorized sche1ne for the sa1ne reason as before. The starting point is to
eli1ninate Yn+i/ 2 from equations ( 46) in the process of transformations