By John P. Boyd
Thoroughly revised textual content makes a speciality of use of spectral how you can remedy boundary worth, eigenvalue, and time-dependent difficulties, but additionally covers Hermite, Laguerre, rational Chebyshev, sinc, and round harmonic capabilities, in addition to cardinal features, linear eigenvalue difficulties, matrix-solving tools, coordinate modifications, tools for unbounded periods, round and cylindrical geometry, and masses extra. 7 Appendices. word list. Bibliography. Index. Over a hundred and sixty textual content figures.
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Extra info for Chebyshev and Fourier spectral methods
13 shows the initial condition (Eq. 42) and its second derivative. The latter has jump discontinuities at x = ±0, π, 2 π, . . At t = 0, these discontinuities cause no problems for a Chebyshev expansion because the Chebyshev series is restricted to x ∈ [0, π] (using Chebyshev polynomials with argument y ≡ (2/π)(x − π/2)). On this interval, the initial second derivative is just the constant −2. For t > 0 but very small, diffusion smooths the step function discontinuities in uxx , replacing the jumps by very narrow boundary layers.
39) (Birkhoff and Lynch, 1984). The singularity is “weak” in the sense that u(x, y) and its first two derivatives are bounded; it is only the third derivative that is infinite in the corners. Constant coefficient, constant forcing, singular solution? It seems a contradiction. However, the boundary curve of a square or any other domain with a corner cannot be represented by a smooth, infinitely differentiable curve. At a right-angled corner, for example, the boundary curve must abruptly shift from vertical to horizontal: the curve is continuous, but its slope has a jump discontinuity.
It is generally impossible to estimate these various errors precisely for the unknown solution to a differential equation. We have two useful alternatives. One is to look at the numerically-computed spectral coefficients, as described in Sec. 12. The other strategy, and the only one which can be applied a priori is defined by the following. Definition 11 (Method of Model Functions) Truncation error is estimated by computing the expansion of a KNOWN function which is “similar”, in a technical sense to be explained later, to the solution of the target differential equation, and then truncating the known expansion after N + 1 terms.