Differential Equations: The First Steps - Elements of Scientific Computing - page 52

explicit (upper), exact, implicit (lower), N = 25, T = 1.0

0

−1

−2

−3

−4

−5

−6

−7

−8

−9

−10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

Fig. 2.3 The figure shows the numerical solution generated by an explicit scheme ( upper curve ),
the analytical solution ( middle ), and the solution generated by an implicit scheme ( lower ). All the
solutions are plotted by drawing a straight line between the computed values. This is also done for
the analytical solution, which is therefore correct only at t 0 ;t 1 t N

In Fig. 2.3 we have plotted the two numerical solutions and the exact solution for t
ranging from 0 to 1: In the numerical computations we used N D 25: We note that
both numerical solutions behave well.
But let us look a bit closer at what happens when we reduce the number of
time steps, i.e., we increase the size of t: In Table 2.1 , we compare the numerical
solution generated by the implicit and explicit scheme at time t D 1:
We see from Table 2.1 that the implicit scheme gives reasonable solutions for
any t; whereas the explicit scheme runs into serious trouble as N becomes smaller
than 11.

The Explicit Scheme

We can see this effect directly from the explicit scheme. Suppose y n <0;and recall
that

y nC1 D y n C ty n

:

Hence, in order for y nC1 to be negative, we must have

y n C ty n <0;