1549301742-The_Theory_of_Difference_Schemes__Samarskii

The sum1narized approxilnation method 599

As a matter of experience, the esti1nation of the nearness of a solution

of the difference proble1n amounts to the proximity between a solution of

the original problem (5) and a solution of the chain of problems (6)-(7).

The main idea behind this approach is connected with the obvious relation

In addition to (6)-(7), the second chain of the equations

( 11)

under the additional conditions of conjugation complements subsequent

studies

(12)

The solution of this problem is the function v(x, t) = v(p)(x, t).
In contrast to (6)-(7), every equation with the number Cl' is solved
here on the whole interval tj < t < tj+! · It is interesting to note that in
some particular cases solutions of problem (6)-(7) and proble1n (11)--(12)
will coincide. This is certainly true in the situation when both .f"' _ U and
Lo: are independent oft.
Along these lines, both chains generate approximations on the solution
u = it(x, t) of the original problem (5). Indeed, it is straightforward to verify
for problem (6)-(7) with the aid of the relation Pu u =(Po: u)j+i/^2 + O(r)
that
0 0
1/Ju = ljJ o: + ~':., where 1/• a= (P,, u)j+l/^2 , 1/,: = O(r),

where 1/Jo: = P 0 u(x, t) is the residual for equation (6) with the number Cl'.

In view of this, it follows from the foregoing that

p p

1/J = L 1/Jo: = L 1/J: = O(r)'

it being understood that the syste1n (6)-(7) approximates equation (5) in
a summarized sense.