Difference approximation of elementary differential operators 77ory.j+1 - (2 + h
2
z-1 T ) y.i+1 i + y.j+1 i+1 = -h2 pi z ' O<i<N,So far we have considered the first kind boundary conditions approx-
imated exactly on grids. In the case of the third kind boundary conditions
the question of their approximation needs investigation. In the next section
we will say a little more about this.- Convergence and accuracy of schemes. While solving a problem by
an approximate method the accuracy which is provided by this method
should be properly evaluated and predicted before proceeding to further
constructions. In this regard, the question of convergence and accuracy of
difference schemes arises naturally.
For convenience in analysis, we look for in a domain G with the bound-
ary r a solution to the linear differential equation
(35) Lu=f(x), x E G,
subject to additional (boundary or initial) conditions(36) lu=ft(x), x E f,
where f(x) and ft(x) are given functions (the input data of the problem)
and l is a linear differential operator, under the agreement that a solution
of problem (35)-(36) exists and is unique.
Within the framework of the present chapter the domain G + r of
continuous variation of an argument (point) is replaced by some discrete
set of points (nodes) X; known as a grid.
Let h be a vector parameter related to the distribution density of the
nodes of the grid and let w h and rh be the sets of its inner and boundary
nodes. With these ingredients, the difference proble111
(37) for x E 'Y 1h'where 'Ph(x) and xh(x) are given grid functions, is put in correspondence
with problem (35)-(36). Here the operators Lh and lh assign the values to
grid functions defined for x E wh = wh + rh. A solution Yh of problem (37)
is a grid function of the nodes of the grid w 1 ,. Varying hand thereby com-
posing different grids w 1 ., we constitute the set of solutions {Yh} depending
on the parameter h. In this context, the family of difference problems (37)