1549301742-The_Theory_of_Difference_Schemes__Samarskii

546 Economical Difference Schemes for Multidimensional Problerns

Because of these facts, tlw number of the necessary operations at every

node of the grid is independent of the total nu1nber of the grid nodes. All

the schemes with the indicated property a.re said to be econo1nical.

In what follows one possible example demonstrates for a. system of

ordinary differential equations that there is an implicit scheme which is

rather economical than the explicit ones requiring the additional operations.

Exa1nple vVith this aim, it seems worthwhile giving the following syste1n

of differential equations:

t > 0, u(O) = u 0 ,

where u = (uCll(t), ... , uCml(t)) is a. vector of order m and A= (a;j) is a.

syn1metric positive definite matrix. In passing from one layer to another the

explicit sche111e Yn+l = Yn -TAYn requires 2m^2 + 21n a.ritlunetic operations.

Furthern1ore, let A - = (a:-) IJ and A+ = (a+) IJ be an upper and a. lower

tridiagonal ma.trices with coinciding n1a.in dia.o·ona.ls ~ a:-. zz = a~ iz = 0 .5 a,., · ..

Both ma.trices (opera.tors) a.re positive definite in the sense of the usual

inner product in the space Rm, since A-= (A+)* and

Before going further, we initiate the construction of the sche111e

(4)

(5) n = 0, 1, ... ,

in which it is necessary to perform the inversion of both tridiagonal matrices

( E + TA-) and ( E +TA+) in determining y 211 + 1 and y 211 + 2.

It is plain to show that the scheme in view is absolutely stable for any

T > 0, permitting one to eliminate y 211 + 1 from the difference equations (4)

and (5). By subtracting equation (5) from equation (4) we find that

Upon substituting the resulting expression into (5) we obtain through such
an analysis the scheme

(6) B _Y2n+~2 __ Y_2_n + A Y2n = ,^0
2T