Heat conduction equation with constant coefficients 301- The family of six-point schemes. As before, it will be convenient to
introduce the grids wh = {xi = ih, i = 0, 1, ... , N} and W 7 = { tj
jr, j = 0, 1, ... , j 0 }. When operating in Don the grid
whr=whxwr={(ih,jr), i=O,l, ... ,N,j=O,l, ... ,j 0 }
with steps h = 1/ N and T = T / j 0 , we denote by yj the value at the
node (xi, tj) of the grid function y given on w hr. The approximation here
consists of replacing the first derivative au./ 8t by the first order difference
derivative, the second derivative 82 u/8x^2 by the second-order difference
derivative uxx =Au and introducing an arbitrary real parameter CJ'. As a
final result we get a one-parameter family of difference schemes
j +1 j
(4) Y; : Y; = A((J'y/+^1 + (1 - CJ')y/) +<pf, 0 < i < N, 0 < j < j 0 •Sometimes scheme ( 4) will be treated as a scheme with weights. The
supplementary boundary and initial conditions will be explicitly specified
with accurate approximations:(5)(6)Y 0 j = uj I (^1) YN j = uj 2)
y~ = y(x;, 0) = u 0 (x;).
Here <p/ is a grid function approximating the right-hand side f of equation
(3). With such a variety, we may accept, for exarnple,
A Yi = Yxx, i = ( Yi-1 - 2 Yi + Yi+1) / h
2
·
When all conditions (4)-(6) are put together, they are constitute problem
(II)..
The difference scheme ( 4) is constructed on the six-point pattern with
the nodes
and center (x;, ij+i) (see Fig. 4.c). The truth of equation (4) is supposed
at the nodes (xi, ij+ 1 ), i = 1,2, ... ,N-l; j + 1=1,2, ... ,j 0 , known as
the inner nodes. The set of all inner nodes of the grid whr is denoted by