Pseudo-spectral methods, also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. They are closely related to spectral methods, but complement the basis by an additional pseudo-spectral basis, which allows representation of functions on a quadrature grid. This simplifies the evaluation of certain operators, and can considerably speed up the calculation when using fast algorithms such as the fast Fourier transform.
Motivation with a concrete example Take the initial-value problem
i ∂ ∂ t ψ ( x , t ) = [ − ∂ 2 ∂ x 2 + V ( x ) ] ψ ( x , t ) , ψ ( t 0 ) = ψ 0 {\displaystyle i{\frac {\partial }{\partial t}}\psi (x,t)={\Bigl [}-{\frac {\partial ^{2}}{\partial x^{2}}}+V(x){\Bigr ]}\psi (x,t),\qquad \qquad \psi (t_{0})=\psi _{0}}
with periodic conditions ψ ( x + 1 , t ) = ψ ( x , t ) {\displaystyle \psi (x+1,t)=\psi (x,t)} . This specific example is the Schrödinger equation for a particle in a potential V ( x ) {\displaystyle V(x)} , but the structure is more general. In many practical partial differential equations, one has a term that involves derivatives (such as a kinetic energy contribution), and a multiplication with a function (for example, a potential). In the spectral method, the solution ψ {\displaystyle \psi } is expanded in a suitable set of basis functions, for example plane waves,
ψ ( x , t ) = 1 2 π ∑ n c n ( t ) e 2 π i n x . {\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\sum _{n}c_{n}(t)e^{2\pi inx}.}
Insertion and equating identical coefficients yields a set of ordinary differential equations for the coefficients,
i d d t c n ( t ) = ( 2 π n ) 2 c n + ∑ k V n − k c k , {\displaystyle i{\frac {d}{dt}}c_{n}(t)=(2\pi n)^{2}c_{n}+\sum _{k}V_{n-k}c_{k},}
where the elements V n − k {\displaystyle V_{n-k}} are calculated through the explicit Fourier-transform
V n − k = ∫ 0 1 V ( x ) e 2 π i ( k − n ) x d x . {\displaystyle V_{n-k}=\int _{0}^{1}V(x)\ e^{2\pi i(k-n)x}dx.}
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