Spectral method

Summary

A spectral method approximates the solution of a differential equation by a finite sum of smooth global functions, such as trigonometric functions or Chebyshev polynomials. The coefficients are fixed by a Galerkin condition, a tau condition or collocation at chosen points. For smooth solutions the error decreases faster than any power of the number of terms, often exponentially. So they are the method of choice for smooth problems on simple domains. Examples are turbulence in periodic boxes and weather models on the sphere. See the definition.