As we have seen, some m-step linear multistep methods assume the knowledge of the approximations at the first m mesh-points of the mesh, i.e., the values y0, y1, . . . , ym−1 are considered as given.
However, from the initial condition (2.2) we know only the value y0. The other initial conditions to the linear multistep methody1, . . . , ym−1 are usually defined by the use of some suitably chosen one-step method. (This means that the accuracy of the one-step method coincides with the accuracy of the applied linear multistep method. Typically, this one-step method is some Runge–Kutta method with sufficiently high accuracy.) We emphasize that the orders of the linear multistep method and the applied one-step method should be equal, because if the one-step method has lower order, we lose the order of accuracy of the linear multistep method.)1
We consider the convergence of linear multistep methods. As we have already seen in the case of one-step methods, consistency in itself is not enough for the convergence. In the sequel (without proof) we give the conditions under which the convergence holds.
For a consistent method the value ξk = 1 is a root of the first characteristic equation. (See condition (3.21).) The following definition tells us what other roots are allowed.
Definition 3.2.1. We say that a linear multistep method satisfies the root criterion when for the roots ξk ∈ C (k = 1,2, . . . , m) of the characteristic equation %(ξ) = 0 the inequality |ξk| ≤1 holds, and the roots with the property
|ξk|= 1 are single.
The next theorem shows that the root criterion for a linear multistep method means its stability.
Theorem 3.2.2. Assume that a linear multistep method is consistent, and the root criterion is valid. Then the method is convergent, i.e., for any fixed point t? ∈(0, T) we have the relation yn→u(t?) as h→0, where nh=t?.
Remark 3.2.1. The following example illustrates the role of the root cri-terion: when it is destroyed, then the method is not stable (and hence, not convergent). We consider the two-step, explicit linear multistep method of the form
yi+ 4yi−1−5yi−2 =h(4fi−1+ 2fi−2).
We can easily check that the method has maximum accuracy, i.e, its order is p = 2m−1 = 3. The first characteristic polynomial is %(ξ) = ξ2+ 4ξ−5 = (ξ − 1)(ξ + 5). The roots are obviously ξ1 = 1 and ξ1 = 5, which means that the root criterion is not satisfied. Let us consider the equation u0 = 0
1We note that in the program package the approximations yk (k = 1,2, . . . m−1) are defined by a suitably chosen k−1-step method.
with the initial condition u(0) = 0. (Clearly, the exact solution is the function u(t) = 0.) We solve numerically this problem using the above method. We choose y0 = 0 and y1 = ε. (When we compute the first approximation y1 by some one-step method, then the result will be a good approximation to the exact solution, which means that it will be close to zero. Hence, ε is a small number, close to zero.) Then the use of the numerical method results in the following approximations:
y2 =−4y1 =−4ε y3 =−4y2+ 5y1 = 21ε
y4 =−4y3+ 5y2 =−104ε etc.
We can observe that the numerical results are increasing, and the values of the approximations are not bounded. Hence, the considered numerical method is not convergent.2
We note that the above example shows that the root criterion is a necessary condition of the convergence. However, the question of its sufficiency is yet to be answered. (I.e., does the root criterion guarantee the convergence?) The answer to this question is negative, i.e., the root criterion in itself is not enough for the convergence. This means that even under the root criterion some numerical methods result in bad approximations due to the errors arising in the computations. In these methods the problem is that their characteristic equation has more than one different roots with absolute values equal to one.
To this aim, we introduce the following notion.
Definition 3.2.3. We say that a linear multistep method is strongly stable, when it satisfies the root criterion, and ξk = 1is the only root with the property
|ξk|= 1.
As an example, we consider the Milne method, having the form yi −yi−2 = h
3(fi+ 4fi−1 +fi−2).
The roots of its characteristic polynomial are ξ1,2 = ±1. Hence, the root criterion is true, but the method is not strongly stable. Therefore, the usage of this method is not recommended. For a strongly stable linear multistep method we recall the famous result, given by G. Dahlquist, which shows that the maximum order of such methods are rather restrictive.
2We emphasize that in this example εcan be considered as some small perturbation of the exact value, i.e., even putting for y1 its exact value u(h), (which is theoretically equal to zero), due to the computer representation, we have only some approximate value, which can be considered as y1=ε.
Theorem 3.2.4. The maximum order of an m-step linear multistep method is p=m+ 1.
For one-step methods we have already shown that the convergence does not give any information about its adequate behavior on some fixed mesh. (The convergence ensures that for sufficiently small mesh-size the numerical result is close to the exact solution. However, as we have seen, this small step size can be unrealistic.) To avoid this situation, we have defined the absolute stable methods. (See Definition 2.4.9.)
For a linear multistep methods the following question is quite natural:
Which linear multistep methods are absolutely stable? The answer to this question shows that for these methods it is difficult to guarantee the absolute stability. Namely, according to the first and second order barrier, given by Dahlquist, we have
• Explicit linear multistep methods are not A-stable.
• The maximum order of an A-stable linear multistep method is two.
These barriers yield a serious problem in the application of linear multistep methods.