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Extra resources for Analysis of the Clinton Critical Experiment [declassified]
In this method, we do not go through the mode space at all. The differentiation matrix takes us from physical space to physical space, and the act of differentiation is hidden in the matrix itself. The computational cost of the matrix method is the cost of a matrix-vector product, which is an O(N 2 ) operation, rather than the cost of O(Nlog(N )) in the method using expansion coefﬁcients. However, the efﬁciency of the FFT is machine dependent and for small values of N it may be faster to perform the matrix-vector product.
Furthermore, we assume that u(x) and its derivatives are smooth enough to allow for any Fourier expansions which may become required. The ﬁrst two sections of this chapter feature the Fourier–Galerkin and Fourier–collocation methods. The ﬁnal section discusses the stability of these methods. , B u N (x, t) = an (t)einx . |n|≤N /2 Note that an (t) are unknown coefﬁcients which will be determined by the method. In general, the coefﬁcients an (t) of the approximation are not equal to the Fourier coefﬁcients uˆ n ; only if we obtain the exact solution of the problem will they be equal.
We begin with the simple observation that, (u, P N v) = (P N u, P N v) + ((I − P N )u, P N v) . 54 Fourier spectral methods ˆ N with The second term is the scalar product of the projection of v on the space B ˆ N . , (u, P N v) = (P N u, P N v). By the same argument, we ﬁnd that (P N u, v) = (P N u, P N v). Therefore, (u, P N v) = (P N u, v), which means that P N = P N∗ . Now, we notice that the Fourier–Galerkin method involves seeking the trigonometric polynomial u N such that ∂u N = P N LP N u N = L N u N .