Definition 15. For a numerically-stable semi-discrete approximation, there are resolution independent
constants such that for all initial data
Similar definitions hold in the fully-discrete case and/or when the spatial approximation is not a finite
difference one. Essentially, (7.115) attempts to capture the notion that the numerical solution should not
have, at a fixed resolution, growth in time, which is not present at the continuum. However,
the problem with the definition is that it is not useful if the estimate (7.113
) is not sharp,
since neither will be the estimate (7.114
), and the numerical solution can still exhibit artificial
growth.
Example 41. Consider the problem (drawn from [278])
Defining
it follows that This energy estimate implies that the spatial norm of If the system (7.116, 7.117
, 7.118
) is approximated by
On the other hand, discretizing the system as
is, in general, not equivalent to (7.122 The right panel of Figure 3 shows a comparison between discretizations (7.122
) and (7.124
), as
well as (7.122
) with the addition of numerical dissipation (see Section 8.5), in all cases at the
same fixed resolution. Even though numerical dissipation does stabilize the spurious growth
in time, the strictly-stable discretization (7.124
) is considerably more accurate. Technically,
according to Definition 15, the approximation (7.122
) is also strictly stable, but it is more useful to
reserve the term to the cases in which the estimate is sharp. The approximation (7.124
), on the
other hand, is (modulo the flux at boundaries, discussed in Section 10) energy preserving or
conservative.
In order to construct conservative or time-stable semi-discrete schemes, one essentially needs to write the approximation by grouping terms in such a way that when deriving at the semi-discrete level what would be the conservation law at the continuum, the need of using the Leibnitz rule is avoided. In addition, the numerical imposition of boundary conditions also plays a role (see Section 10).
In many application areas, conservation or time-stability play an important role in the design of
numerical schemes. That is not so much (at least so far) the case for numerical solutions of Einstein’s
equations, because in general relativity there is no gauge-invariant local notion of conserved energy unlike
many other nonlinear hyperbolic systems (most notably, in Newtonian or special relativistic Computational
Fluid Dynamics); see, however, [400]. In addition, there are no generic sharp estimates for the growth of the
solution that can guide the design of numerical schemes. However, in simpler settings such as fields
propagating on some stationary fixed-background geometry, there is a notion of conserved local energy and
accurate conservative schemes are possible. Interestingly, in several cases such as Klein–Gordon or
Maxwell fields in stationary background spacetimes the resulting conservation of the semi-discrete
approximations follows regardless of the constraints being satisfied (see, for example, [278]). A local
conservation law in stationary spacetimes can also guide the construction of schemes to guarantee
stability in the presence of coordinate singularities [105
, 375
, 225
, 310], as discussed in Section
7.6.
In addition, there has been work done on variational, symplectic or mimetic integration techniques for Einstein’s equations, which aim at exactly or approximately preserving the discrete constraints, while solving the discrete evolution equations. See, for example, [304, 139, 201, 200, 76, 110, 359, 173, 358, 174, 360, 357].
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Living Rev. Relativity 15, (2012), 9
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