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[definition needed].
This specific example is the Schrödinger equation for a particle in a potential
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).
is expanded in a suitable set of basis functions, for example plane waves, Insertion and equating identical coefficients yields a set of ordinary differential equations for the coefficients, where the elements
are calculated through the explicit Fourier-transform The solution would then be obtained by truncating the expansion to
For the numerical solutions, the right-hand side of the ordinary differential equation has to be evaluated repeatedly at different time steps.
At this point, the spectral method has a major problem with the potential term
need to be evaluated explicitly before the differential equation for the coefficients can be solved, which requires an additional step.
In the pseudo-spectral method, this term is evaluated differently.
, an inverse discrete Fourier transform yields the value of the function
However, the pseudo-spectral method allows the use of a fast Fourier transform, which scales as
In a more abstract way, the pseudo-spectral method deals with the multiplication of two functions
The coefficients are then obtained by A bit of calculus yields then with
basis functions, one can try to find a quadrature, i.e., a set of
points and weights such that Special examples are the Gaussian quadrature for polynomials and the Discrete Fourier Transform for plane waves.
The quadrature allows an alternative numerical representation of the function
is then done at each grid point, This generally introduces an additional approximation.
The pseudo-spectral method thus introduces the additional approximation If the product
can be represented with the given finite set of basis functions, the above equation is exact due to the chosen quadrature.
are imposed on the system, the basis functions can be generated by plane waves, with
basis functions, the pseudo-spectral method gives accurate results if
An expansion in plane waves often has a poor quality and needs many basis functions to converge.
As a consequence, plane waves are one of the most common expansion that is encountered with pseudo-spectral methods.
Here, the Gaussian quadrature is used, which states that one can always find weights
are chosen for a specific problem, and leads to one of the different forms of the quadrature.
To apply this to the pseudo-spectral method, we choose basis functions
form an orthonormal basis with respect to the scalar product
This basis, together with the quadrature points can then be used for the pseudo-spectral method.
Such polynomials occur naturally in several standard problems.
For example, the quantum harmonic oscillator is ideally expanded in Hermite polynomials, and Jacobi-polynomials can be used to define the associated Legendre functions typically appearing in rotational problems.