Spectral method

A numerical method that approximates a function or differential-equation solution with a finite sum of global basis functions. For smooth problems, its error can decrease faster than any fixed power of resolution.

Fourier spectral method: A spectral method that expands periodic functions in Fourier modes. Fourier modes provide the global basis for periodic problems.
Basis function: A function in a set whose linear combinations represent elements of a function space. Spectral approximations are finite combinations of such functions.
Computational fluid dynamics: The numerical simulation of fluid motion using discretized governing equations. Spectral discretizations are used when fluid solutions are smooth and high accuracy matters.
Finite difference method: A numerical method that approximates derivatives using function values at nearby grid points. Unlike spectral differentiation, it typically uses local stencils and has algebraic accuracy.
Gibbs phenomenon: Persistent oscillatory overshoot near a jump discontinuity in truncated Fourier or related expansions. Global expansions develop oscillations near shocks even as resolution increases.
Chebyshev spectral method: A spectral method using Chebyshev polynomials as basis functions or interpolation nodes. Chebyshev expansions handle nonperiodic intervals without imposing periodicity.
Fourier series: A representation of a periodic function as a sum of sine and cosine waves or complex exponentials. It supplies the standard basis for periodic spectral approximations.
Direct numerical simulation: A computational approach that resolves the full range of dynamically relevant scales in fluid turbulence. Spectral methods are common in simulations of turbulence in periodic domains.
Finite element method: A numerical method that approximates solutions with piecewise functions over a mesh. Its local elements accommodate complex geometries more naturally than global spectral bases.
Runge phenomenon: Large oscillations that can arise when high-degree polynomial interpolation uses unsuitable, typically equally spaced, nodes. Poor node choices can undermine polynomial spectral approximations despite smooth data.

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