- Pseudo-spectral method
Pseudo-spectral methods are a class of
numerical methodsused in applied mathematicsand scientific computingfor the solution of PDEs, such as the direct simulation of a particle with an arbitrary wavefunctioninteracting with an arbitrary potential. They are related to spectral methods and are used extensively in computational fluid dynamics and other areas, but are demonstrated below on an example from quantum physics.
The Schrödinger wave equation,
can be written
which resembles the
linear ordinary differential equation
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 .
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. 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(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.
The error in the pseudo-spectral method is overwhelmingly due to
Wikimedia Foundation. 2010.
Look at other dictionaries:
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… … Wikipedia
Least-squares spectral analysis — (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. [cite book | title = Variable Stars As Essential Astrophysical Tools | author = Cafer Ibanoglu |… … Wikipedia
Split-step method — In numerical analysis, the split step (Fourier) method is a pseudo spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies … Wikipedia
Monte Carlo method — Not to be confused with Monte Carlo algorithm. Computational physics … Wikipedia
List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia
List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra … Wikipedia
Computational physics — This article is about computational science applied in physics. For theories comparing the universe to a computer, see digital physics. Computational physics … Wikipedia
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… … Wikipedia
Computational electromagnetics — Computational electromagnetics, computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computationally… … Wikipedia
Mathematics of general relativity — For a generally accessible and less technical introduction to the topic, see Introduction to mathematics of general relativity. General relativity Introduction Mathematical formulation Resources … Wikipedia