... components^1.1

In the nonlinear case, however, one might argue that the use of higher sampling rates is justifiable, due to the possibility of aliasing. On the other hand, in all physical systems, loss becomes extremely large at high frequencies, so a more sound, and certainly much more computationally efficient approach is to introduce such losses into the model itself. Another argument for using an elevated sample rate, employed by many authors, is that numerical dispersion (leading to potentially audible distortion) may be reduced; this, however, is disastrous in terms of computational complexity, as the total operation count often scales with the square or cube of the sample rate. It is nearly always possible to design a scheme with much better dispersion characteristics, which still operates at a reasonable sample rate.

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... same^1.2

It is possible, for certain systems such as the ideal membrane, under certain conditions, to extract groups of harmonic components from a highly inharmonic spectrum, and deal with them individually using waveguides [5,26], leading to an efficiency gain, albeit a much more modest one than in the 1D case. Such techniques, unfortunately, are rather restrictive in that only extremely regular geometries may be dealt with.

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... fixed^3.1

It is also worth mentioning that in a distributed setting, involving approximations in both space and time, the spatial temporal warping errors often tend to cancel one another to a certain degree (indeed, perfectly, in the case of the 1D wave equation)

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... domain^6.1

One of the nice features of energy analysis is that it allows boundary conditions at different locations to be analyzed independently; this is a manifestly physical approach, in that one would not reasonably expect boundary conditions at separate locations to interact, energetically. The same simplicity of analysis follows for numerical methods, as will be seen in the first instance later in this chapter.

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... methods^13.1

It is worth being quite clear on the use of the word ``spectral," in this case, which is not used in the sense of spectral modeling. The word spectral refers to extremely high accuracy of numerical methods--though they can be based on the use of Fourier series, in most cases of interest they are rather generally based on more general polynomial series approximations.

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