By John P. Boyd
Thoroughly revised textual content specializes in use of spectral easy methods to clear up boundary price, 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 adjustments, equipment for unbounded periods, round and cylindrical geometry, and masses extra. 7 Appendices. thesaurus. Bibliography. Index. Over one hundred sixty textual content figures.
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Additional resources for Chebyshev and Fourier spectral methods
LOCATION OF SINGULARITIES 39 condition which is spatially periodic and antisymmetric with respect to both the origin and x = π. 43) where the sign function equals one when its argument is positive and is equal to minus one when its argument is negative. 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)).
The reason is that the timedependence can be marched forward, from one time level to another. Marching is much cheaper than computing the solution simultaneously over all space-time. A space-only spectral discretization reduces the original partial differential equation to a set of ordinary differential equations in time, which can then be integrated by one’s favorite Runge-Kutta or other ODE time-marching scheme. 28) with the boundary conditions that the solution must be periodic with a period of 2π.
The series with only the constant and the cosine terms is known as a “Fourier cosine series”. ) If f (x) = −f (−x) for all x, then f (x) is said to be antisymmetric about x = 0 and all the an = 0. Its Fourier series is a sine series. These special cases are extremely important in applications as discussed in the Chapter 8. 2. 6) f (x) = f (x + 2π) for all x. To illustrate these abstract concepts, we will look at four explicit examples. These will allow us to develop an important theme: The smoother the function, more rapidly its spectral coefficients converge.