Stability of a difference scheme 97which is approximated on a grid wh = wh +In by a difference problem
(20) L1i Yh ='Ph for x E wh, lh Yh =[th for i: E "ih.
The statement of the problem for the error zh = Yh - uh, where uh 1s
the projection of the solution u of problem (19) onto the grid wh, is
(21) Lhzh=i/Jh for xEwh, lhzh=vh for XE/h,
where 1/Jh and vh are the errors of approximation to the governing equation
and the additional condition. Instead of (21) we may write down formally- Lh zh = 1/Jh ·
If the operator Lh is linear and the difference scheme is well-posed, we are
led by (18) to
II zh ll(lh) < M II ~h llczh) or
II zh ll(lh) < M (111/Jh llc2h) +II vh l/(3h)) ·
(22)vVith these relations established, we conclude that if the scherne is stable
and approximates the original problem, then it is convergent. In other
words, "convergence follows from approximation and stability" and the
order of accuracy and the rate of convergence are connected with the order
of approxin1ation.
Everything just said means that in establishing convergence and in
determining the order of accuracy of a scheme it is necessary to evaluate
the error of approxi111ation, discover stability and then derive estimates of
the form (22) known as a priori estimates.
Being the elements of different spaces, the solution zh and the right-
hand side 1/Jh of the difference problem should be evaluated in different
norms.
The examples of suitable norms are available in the preceding sec-
tions in which the errors of solutions and approximations on the grid wh
are investigated in the usual way. Unfortunately, it is impossible to get
estimates of the form (22) directly for stability of concrete difference prob-
lems. Auxiliary mathematical tools and techniques such as the summation
by parts formulae, the difference Green formulae and elementary analogs of
embedding theorems are aimed at deriving various estimates for solutions of
difference analogs of boundary-value problems associated with an ordinary
second-order differential equation. In this particular case we will encounter
typical situations related to stability, approximation and accuracy of dif-
ference schemes. Later we will elaborate on this for rather complicated
problems. At the san1e time, subsequent discussions of them will involve
issues which are different fron1 those occurred before.