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A spectral method is a method of numerically solving differential equations by approximating the equations using, for example, a Fourier series, and solving each term. Typically a Fourier transform is used.
A typically-used fourier series describes a function over a finite interval (or a function that repeats over such an interval) as a potentially infinite series of sine functions at various frequencies and phases. If a finite subset of these sine functions sums to a good approximation of the function, then differentials of this set of sine functions may sum to a good approximation of differentials of the function in question.