Definition 11. An approximation-discretization to the Cauchy problem (7.1, 7.2) is numerically stable if there is some discrete norm in space and constants such that the corresponding approximation satisfiesfor high enough resolutions, smooth initial data , and .
In other words, the addition of a lower-order term does not affect numerical stability, and without loss of generality one can restrict stability analyses to the homogeneous case.
From here on denotes some discretization of space and time. This includes both finite difference and spectral collocation methods, which are the ones discussed in Sections 8 and 9, respectively. In addition, we use the shorthand notation
In order to gain some intuition into the general problem of numerical stability we start with some examples of simple, low-order approximations for a test problem. Consider uniform grids both in space and time
Example 33. The one-sided Euler scheme.
Eq. (7.10) is discretized with a one-sided FD approximation for the spatial derivative and evolved in time with the forward Euler scheme,
Using Parseval’s identity, we findCFL limit, and (7.18) as the von Neumann condition. If , , while for the scheme is unconditionally unstable even though the underlying continuum problem is well posed.
Next we consider a scheme very similar to the previous one, but which turns out to be unconditionally unstable for , regardless of the direction of propagation.
Example 34. A centered Euler scheme.
Consider first the semi-discrete approximation to Eq. (7.10),
The semi-discrete centered approximation (7.20) and the fully-discrete centered Euler scheme (7.21) constitute the simplest example of an approximation, which is not fully-discrete stable, even though its semi-discrete version is. This is related to the fact that the Euler time integration is not locally stable, as discussed in Section 7.3.2.
The previous two examples were one-step methods, where can be computed in terms of . The following is an example of a two-step method.
Example 35. Leap-frog.
A way to stabilize the centered Euler scheme is by approximating the time derivative by a centered difference instead of a forward, one-sided operator:
In the above example the amplification matrix can be diagonalized through a transformation that is uniformly bounded:
The previous examples were explicit methods, where the solution (or ) can be explicitly computed from the one at the previous timestep, without inverting any matrices.
Example 36. Crank–Nicholson.
Approximating Eq. (7.10) by
Example 37. Iterated Crank–Nicholson.
Approximating the Crank–Nicholson scheme through an iterative scheme with a fixed number of iterations is usually referred to as the Iterated Crank–Nicholson (ICN) method. For Eq. (7.10) it proceeds as follows :
The resulting discretization is numerically stable for and iterations, and unconditionally unstable otherwise. In the limit the ICN scheme becomes the implicit, unconditionally-stable Crank–Nicholson scheme of the previous example. For any fixed number of iterations, though, the method is explicit and stability is contingent on the CFL condition . The method is unconditionally unstable for because the limit of the amplification factor approaching one in absolute value [cf. Eq. (7.34)] as increases is not monotonic. See  for details and  for a similar analysis for “theta” schemes.
Similar definitions to the one of Definition 11 are introduced for the IBVP. For simplicity we explicitly discuss the semi-discrete case. In analogy with the definition of a strongly–well-posed IBVP (Definition 9) one has
Definition 12. A semi-discrete approximation to the linearized version of the IBVP (5.1, 5.2, 5.3) is numerically stable if there are discrete norms at and at and constants and such that for high-enough resolution the corresponding approximation satisfiesfor all . If the constant can be chosen strictly positive, the problem is called strongly stable.
In addition, the semi-discrete version of Definitions 6 and 7 lead to the concepts of strong stability in the generalized sense and boundary stability, respectively, which we do not write down explicitly here. The definitions for the fully-discrete case are similar, with time integrals such as those in Eq. (7.39) replaced by discrete sums.
Living Rev. Relativity 15, (2012), 9
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