Pseudo-spectral methods are a class of numerical methods used in applied mathematics

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

 and scientific computing for the solution of PDEs

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

, such as the direct simulation of a particle with an arbitrary wavefunction

Wavefunction

Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

 interacting with an arbitrary potential

Potential energy

In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

. They are related to spectral method

Spectral method

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential...

s and are used extensively in computational fluid dynamics and other areas, but are demonstrated below on an example from quantum physics.

Background

The Schrödinger wave equation,

can be written

which resembles the linear

Linear

In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

 ordinary differential equation

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

with solution

In fact, using the theory of linear operators, it can be shown that the general solution to the Schrödinger wave equation is

where exponentiation of operators is defined using power series. Now remember that

where the kinetic energy

is given by

and the potential energy often depends only on position
(i.e.,
). We can write

It is tempting to write

so that we may treat each factor separately. However, this is only true if the operators and commute, which is not true in general. Luckily, it turns out that

is a good approximation for small values of . This is known as the symmetric decomposition. The heart of the pseudo-spectral method is using this approximation iteratively to calculate the wavefunction for arbitrary values of .

The method

For simplicity, we will consider the one-dimensional case. The method is readily extended to multiple dimensions.

Given , we wish to find where is small. The first step is to calculate an intermediate value by applying the rightmost operator in the symmetric decomposition,

This requires only a pointwise multiplication. The next step is to apply the middle operator,

This is an infeasible calculation to make in configuration space

Configuration space

- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...

. Fortunately, in momentum space, the calculation is greatly simplified. If is the momentum space representation of , then

which also requires only a pointwise multiplication. Numerically, is obtained from using the Fast Fourier transform

Fast Fourier transform

A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

 (FFT) and is obtained from using the inverse FFT.

The final calculation is

This sequence can be summarized as

Analysis of algorithm

If the wavefunction is approximated by its value at distinct points, each iteration requires 3 pointwise multiplications, one FFT, and one inverse FFT. The pointwise multiplications each require effort, and the FFT and inverse FFT each require effort. The total computational effort is therefore determined largely by the FFT steps, so it is imperative to use an efficient (and accurate) implementation of the FFT. Fortunately, many are freely available.

Error analysis

The error in the pseudo-spectral method is overwhelmingly due to discretization error

Discretization error

In numerical analysis, computational physics, and simulation, discretization error is error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a lattice...

.