610 Economical Difference Schemes for Multidimensional Problems- Stability of LOS. The main goal of stability consideration is to establish
that the unifonn convergence with the rate 0( T + lhl^2 ) follows from a
summarized approximation obtained. This can be done using the maximun1
principle and a priori estimates in the grid norn1 of the space C for a solution
of problem (21)-(23) expressing the stability of the scheme concerned with
respect to the initial data, the right-hand side and boundary conditions.
Recall that in Chapter 4, Section 2 we have proved the maximum
principle and derived a priori estimates for a solution to the grid equation
of the general form
(28) A(P) y(P) = B(P, Q) y(Q) + F(P) for PE fl,
QEIII'(P)y(P) = μ(P) for P E S,
where P and Q are smne nodes of a connected grid fl + S and I I I' ( P) is
a neighborhood of the node P except the point P itself. The coefficients
A(P) and B(P, Q) must satisfy the conditions(29) A(P)>O, B(P,Q) > 0
D(P) = A(P) - B ( P, Q) :::: 0.
QEII I'(P)Applying theorems of Section 2, Chapter 4 to proble1n (21)-(23) yields
the following result.
Theoren1 1 The locally one-cli1nensional sche1ne (21)-(23) is uniformly
stable in the metric of the .space C with respect to the initial data. the
right-hand side and boundary conditions and a. solution of problem (21)-
(23) admits for any T and h the estin1ate
(30) Iii/ lie ~ lluo lie+ max. llμ(x, t')lle
O<t'':S:JT "Y
j-1 p
+ O<t'<jT 1nax h^2 ll<p(x, t')lle· +""""' L._, T""""' L._, ll'Pj'+o:/p o II C' a I- -· j'=O o:=l