In general, this method provides only a subclass of numerically-stable approximations. However, it is a very practical one, since spatial and time stability are analyzed separately and stable semi-discrete approximations and appropriate time integrators can then be combined at will, leading to modularity in implementations.
Consider the approximation
In the time-independent case, the solution to (7.76, 7.77) is
Semi-discrete stability also follows if is semi-bounded, that is, there is a constant independent of resolution such that (cf. Eq. (3.25) in Theorem 1)
For a large class of problems, which can be shown to be well posed using the energy estimate, one can construct semi-bounded operators by satisfying the discrete counterpart of the properties of the differential operator in Eq. (7.1) that were used to show well-posedness. This leads to the construction of spatial differential approximations satisfying the summation by parts property, discussed in Sections 8.3 and 9.4.
Now we consider explicit time integration for systems of the form (7.76, 7.77) with time-independent coefficients. That is, if there are points in space we consider the system of ordinary differential equations (ODEs)
In the previous Section 7.3.1 we derived necessary conditions for semi-discrete stability of such systems. Namely, the von Neumann one in its weak (7.80) and strong (7.82) forms. Below we shall derive necessary conditions for fully-discrete stability for a large class of time integration methods, including Runge–Kutta ones. Upon time discretization, stability analyses of (7.85) require the introduction of the notion of the region of absolute stability of ODE solvers. Part of the subtlety in the stability analysis of fully-discrete systems is that the size of the system of ODEs is not fixed; instead, it depends on the spatial resolution. However, the obtained necessary conditions for fully-discrete stability will also turn out to be sufficient when combined with additional assumptions. We will also discuss sufficient conditions for fully-discrete stability using the energy method.
Suppose now that the system of ODEs (7.85) is evolved in time using a one-step explicit scheme,
The necessary condition (7.92) can then be restated as:
Lemma 6 (Fully-discrete von Neumann condition for the method of lines.). Consider the semi-discrete system (7.85) and a one-step explicit time discretization (7.87) satisfying the assumptions (7.89). Then, a necessary condition for fully-discrete stability is that the spectrum of the scaled spatial approximation is contained in the region of absolute stability of the ODE solver ,for all spatial resolutions .
In the absence of lower-order terms and under the already assumed conditions (7.89) the strong von Neumann condition (7.82) then implies that must overlap the half complex plane . In particular, this is guaranteed by locally-stable schemes, defined as follows.
As usual, the von Neumann condition is not sufficient for numerical stability and we now discuss an example, drawn from , showing that the particular version of Lemma 6 is not either.
Example 40. Consider the following advection problem with boundaries:
The corresponding semi-discrete scheme can be written in the form
Since is triangular, its eigenvalues are the elements of the diagonal; namely, , i.e., there is a single, degenerate eigenvalue .
The region of local stability of the Euler method is
Theorem 11 (Kreiss–Wu ). Assume that
Then the fully-discrete system is numerically stable, also in the generalized sense, under the CFL condition
Since the exact solution to Eq. (7.85) is and, in particular,7.106) must agree with the first terms of the Taylor expansion of , where is the order of the global 28 truncation error of the ODE solver, defined through 7.109). In particular, for , such a scheme corresponds to the Euler one discussed in the example above; see Eq. (7.99).
Using the energy method, fully-discrete stability can be shown (resulting in a more restrictive CFL limit) for third-order Runge–Kutta integration and arbitrary dissipative operators [282, 409]:
Theorem 12 (Levermore). Suppose is dissipative, that is Eq. (7.83) with holds. Then, the third-order Runge–Kutta approximation , wherefor the semi-discrete problem (7.85) is strongly stable, under the CFL timestep restriction .
Notice that the restriction is not so severe, since one can always achieve it by replacing with . A generalization of Theorem 12 to higher-order Runge–Kutta methods does not seem to be known.
Living Rev. Relativity 15, (2012), 9
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