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

. This specific example is the Schrödinger equation, 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).

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,

are calculated through the explicit Fourier-transform

. In general, this is done by numerical methods, such as Runge–Kutta methods.

need to be evaluated explicitly before the differential equation for the coefficients can be solved, which requires an additional step.

.

can be used directly without evaluating any additional integrals.

Technical discussion

as part of a partial differential equation. To simplify the notation, the time-dependence is dropped. Conceptually, it consists of three steps:

1. are expanded in a finite set of basis functions (this is the spectral method).

2. For a given set of basis functions, a quadrature is sought that converts scalar products of these basis functions into a weighted sum over grid points.

3. at each grid point.

Expansion in a basis

as

using the inner product. The coefficients are then obtained by

A bit of calculus yields then

from the quadrature basis, the expansion is sometimes called Finite Basis Representation (FBR).

Quadrature

points and weights such that

Special examples are the Gaussian quadrature for polynomials and the Discrete Fourier Transform are a function of the basis and.

through their value at the grid points. This representation is sometimes denoted Discrete Variable Representation (DVR), and is completely equivalent to the expansion in the basis.

Multiplication

is then done at each grid point,

:

. The pseudo-spectral method thus introduces the additional approximation

can be represented with the given finite set of basis functions, the above equation is exact due to the chosen quadrature.

Special pseudospectral schemes

The Fourier method

are imposed on the system, the basis functions can be generated by plane waves,

is the ceiling function.

is given by the discrete Fourier transformation.

For the discussion of the error, note that the product of two plane waves is again a plane wave basis functions are used.

An expansion in plane waves often has a poor quality and needs many basis functions to converge. However, the transformation between the basis expansion and the grid representation can be done using a Fast Fourier transform. As a consequence, plane waves are one of the most common expansion that is encountered with pseudo-spectral methods.

Polynomials

Another common expansion is into classical polynomials. Here, the Gaussian quadrature such that

or less. Typically, the weight function with the property

. This basis, together with the quadrature points can then be used for the pseudo-spectral method.

basis functions, and the pseudo-spectral method will give accurate results for that many basis functions.

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.

See also

• Pseudospectral time-domain method

Notes

References

• Jie Shen, Tao Tang and Li-Lian Wang (2011) "Spectral Methods: Algorithms, Analysis and Applications" (Springer Series in Computational Mathematics, V. 41, Springer), .