1 | More precisely, it follows from the Paley–Wiener theorem (see Theorem IX.11 in [346]) that ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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2 | In this regard we should mention the Cauchy–Kovalevskaya theorem (see, for example, [161]), which always provides unique local analytic solutions to the Cauchy problem for quasilinear partial differential equations with analytic coefficients and data on a non-characteristic surface. However, this theorem does not say anything about causal propagation and stability with respect to high-frequency perturbations. | |
3 | In fact, we will see in Section 3.2 that in the variable coefficient case, smoothness of the symmetrizer in ![]() |
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4 | Here, the factor ![]() ![]() ![]() ![]() |
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5 | Notice that ![]() ![]() ![]() ![]() ![]() ![]() |
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6 | Here, the advection term ![]() ![]() |
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7 | The Fourier transform of the function ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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8 | These smoothness requirements are sometimes omitted in the numerical-relativity literature. | |
9 | In principle, the maximum propagation speed ![]() ![]() ![]() ![]() ![]() |
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10 | See Theorem 4.1.3 in [327![]() |
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11 | Geometrically, this means that the identity map ![]() ![]() ![]() ![]() ![]() ![]() |
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12 | As indicated above, given initial data ![]() ![]() ![]() ![]() ![]() ![]() |
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13 | Notice that the condition of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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14 | Weak hyperbolicity of the system (4.20![]() ![]() ![]() ![]() ![]() |
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15 | Notice that even when ![]() ![]() ![]() ![]() ![]() ![]() |
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16 | If ![]() ![]() |
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17 | However, one should mention that this convergence is usually not sufficient for obtaining accurate solutions. If the
constraint manifold is unstable, small departures from it may grow exponentially in time and even though
the constraint errors converge to zero they remain large for finite resolutions; see [254![]() |
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18 | However, it should be noted that these solutions are not square integrable, due to their harmonic dependency in ![]() ![]() ![]() |
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19 | Alternatively, if ![]() ![]() |
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20 | ![]() ![]() ![]() ![]() |
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21 | One can always assume that ![]() ![]() ![]() ![]() ![]() |
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22 | The restriction to homogeneous boundary conditions ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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23 | Instead of imposing the constraint itself on the boundary one might try to set some linear combination of its normal and
time derivatives to zero, obtaining a constraint-preserving boundary condition that does not involve zero speed fields.
Unfortunately, this trick only seems to work for reflecting boundaries; see [405![]() ![]() ![]() ![]() |
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24 | This relation is explicitly given in terms of the Weyl tensor ![]() ![]() ![]() ![]() ![]() ![]() |
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25 | A systematic derivation of exact solutions including the correction terms in ![]() ![]() ![]() |
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26 | Notice that this is one of the requirements of Theorem 7, for instance, where the boundary matrix must have constant rank in order for the theorem to be applicable. | |
27 | See, for instance, Example 6 with initial data ![]() |
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28 | This refers to the truncation error at a fixed final time, as opposed to the local one after an iteration. | |
29 | There are cases in which this is not true, at least for not-too-large penalty strengths. |
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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