 Spectral method

A spectral method approximates an unknown function by a finite expansion in global basis functions, such as Fourier modes or orthogonal polynomials. A Galerkin method makes the residual orthogonal to the retained trial space; a collocation method instead makes it vanish at selected nodes. The expansion coefficients are the numerical unknowns.

For the $2\pi$-periodic heat equation, a Fourier series truncationgives $a_m^{\prime }(t)=-m^2{a}_m(t)$ under Fourier–Galerkin method projection. Each retained mode decays independently, and differentiation becomes multiplication of its coefficient by $im$. This turns a partial differential equation into finitely many ordinary differential equations.

Rapid convergence comes from rapid coefficient decay. If $|a_m|\le C{e}^{-\sigma |m|}$ with $\sigma >0$, the omitted tail has an exponentially small uniform norm bound by a geometric series. Limited smoothness or discontinuities can prevent that rate. A large number of retained modes cannot justify exponential convergence without the needed solution regularity.

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.