Spectral Methods
Spectral methods expand the solution in global smooth basis functions, achieving exponential accuracy for smooth problems.
Global basis, exponential convergence
Spectral methods represent the solution as a sum of globally defined, infinitely smooth basis functions: Fourier modes on periodic domains, Chebyshev or Legendre polynomials on bounded ones. Because each basis function spans the whole domain, information at any point influences the whole expansion. For analytic solutions the approximation error decays faster than any power of the number of modes, a property called spectral or exponential convergence.
This is dramatically more efficient than finite differences for smooth problems: doubling the modes can square the accuracy rather than merely halving the error. The price is that the basis assumes smoothness; discontinuities produce Gibbs oscillations that spoil accuracy globally.
Galerkin and tau formulations
A spectral Galerkin method requires the residual to be orthogonal to the basis, leading to relations among the expansion coefficients. For Fourier bases, differentiation becomes multiplication by the wavenumber, so linear constant-coefficient operators are diagonal. Boundary conditions on non-periodic domains are handled by the tau method or by choosing basis combinations that satisfy the conditions individually.
When to use them
Spectral methods excel for homogeneous turbulence, periodic transport problems, and high-accuracy reference solutions. Their global coupling makes them harder to parallelize and less flexible in complex geometry than local methods, which motivates the spectral element hybrid.
- Fourier bases for periodic domains, trivial differentiation
- Chebyshev/Legendre bases for bounded domains
- Exponential convergence for analytic solutions
- Gibbs phenomenon near discontinuities limits use for shocks
In fusion turbulence studies, spectral and pseudo-spectral discretizations resolve a wide range of scales efficiently, which is why they appear in gyrokinetic and fluid turbulence codes on periodic or flux-tube domains.