A spectral method approximates a differential equation in a global basis and enforces the equation after projection or collocation. Smooth and analytic solutions often give rapidly convergent coefficients.
A Fourier spectral method represents a periodic unknown by finitely many Fourier modes and converts differentiation into multiplication by wavenumber.
A spectral truncation retains a finite set of basis modes and sets all omitted coefficients to zero.
The Fourier–Galerkin method seeks a finite Fourier series whose residual is orthogonal to every retained Fourier mode.
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Spectral methods are a class of numerical techniques used to solve differential equations by expanding the solution in terms of a set of basis functions. These methods are particularly powerful for solving problems in fluid dynamics, wave propagation, and other areas of physics and engineering. Spectral methods leverage the properties of Fourier series or orthogonal polynomials to achieve high accuracy with relatively few degrees of freedom.