302 Difference Sche1nes with Constant CoefficientsThe boundary and initial conditions (5) and (6) are specified at the
boundary nodes of the grid w hr. All of the nodes of the grid w hr lying
on the straight line t = tj constitute what is called a layer. Because in
scheme (4) knowledge of the values of the sought function y is required on
two layers, it can be regarded as a two-layer scheme. In the sequel we
will show that the accuracy and stability of scheme ( 4) depend on a proper
choice of the parameter (J". Before proceeding to further careful analysis,
it would be prudent to examine some schemes relating to particular values
of (J" in light of those remarks. For (J" = 0 we get the four-point scheme
(Fig. 4.a)
yj+l - y/
T Ay/ +'PI
or(7)developed on the pattern with the nodes (x;, ij+ 1 ), (x;, tj) and (x;± 1 , tj)·
The value y/+^1 at every point of the new layer t = ij+ 1 is given by the
explicit formula (7) through the values y/ on the previous layer t = tj. Since
the solution is prescribed by the initial condition y,,^0 = u 0 (x;) at moment
t = t 0 , it is possible to detern1ine all the values of yon any adjacent layer
by applying successively formula (7). Because of this, scherne (7) is said to
be explicit.
For (J" -::j:. 0 scheme ( 4) refers to an implicit two-layer scheme. When
the value y/ +i is sought on the new layer under the natural premise (J" -::j:. 0,
we obtain the governing system of algebraic equations
. 1. '.
(S) FJ i = -yJ T z +(1-(J")AyJ z +in? ri)
i = 1, 2, ... _, N - 1,
subject to the boundary conditions Yi +^1 = uj +^1 and yJ./^1 = ui +^1 • This
system can be solved by the right elimination method (see Chapter 1, Sec-
tion 2.5). In conclusion, it is worth inentioning two other schemes. The
first one with (J" = 1 known as the forward difference scheme or pure
implicit scheme will appear as further developments occur:
y/+1 - y/
T(9) A y.J+z^1 + 1nj. r,