410 Chapter 7 Numerical Methods
i
m 01234
0 00 .51 0. 50
1 0 0. 50. 50. 5 0
2 0 0000
3 0 − 0. 5 − 0. 5 − 0. 5 0
4 0 − 0. 5 − 1 − 0. 5 0
5 0 − 0. 5 − 0. 50. 5 0
6 0 0000
Table 6 Numerical solution of Eqs. (1)–(3)velocity were not identically zero, the numerical solution would in general be
only an approximation to the true solution.
Stability
In our study of the heat equation, Section 7.2, we saw that the choice of xand
twas not free. The same is true for the wave equation. Suppose we attempt
to solve the same problem as earlier, but withρ^2 =( t/ x)^2 chosen to be 2.
Then Eq. (4) becomes
ui(m+ 1 )= 2(
ui− 1 (m)−ui(m)+ui+ 1 (m))
−ui(m− 1 ),and the “solution” corresponding to this rule of calculation is shown in Table 7
(again, entries in italics are given data). Of course, the results bear no resem-
blance to the solution of the wave equation. They suffer from the same sort of
instability as that observed in Section 7.2. There is a rule of thumb, similar to
theonetobefoundthere,applicabletothewaveequation.
First, write out the equations for eachui(m+ 1 )in terms of theu’s at time
levelsmandm−1:
ui(m+ 1 )=aiui− 1 (m)+biui(m)+ciui+ 1 (m)−ui(m− 1 ).The coefficients must satisfy two conditions:
- None of the coefficientsai,bi,cimay be negative.
- The sum of the coefficients is not greater than 2:
ai+bi+ci≤ 2.Of course,ui(m− 1 )appears with a coefficient of−1; nothing can be done
about that, nor does it enter into the aforementioned rules.