Computational Physics

(Rick Simeone) #1

578 Appendix A


Bulirsch–Stoer method

This method is similar to the Romberg method for numerical integration. Suppose
the value ofxtis wanted,x 0 being known. Over the interval[0,t], the equation can
be integrated using a simple method with a few steps. Then the method is repeated
with a larger number of steps and so on. The resulting predictions forxtare stored
as a function of the inverse of the number of steps used. Then these values are used
to build an interpolation polynomial which is extrapolated to an infinite number
of steps. The efficiency of this method is another reason why predictor-corrector
methods are seldom used.


A7.2 Partial differential equations

In mathematics, one usually classifies partial differential equations (PDE) accord-
ing to their characteristics, leading to three different types: parabolic, elliptic and
hyperbolic. In numerical analysis, this classification is less relevant, and only two
different types of equations are distinguished: initial value and boundary value prob-
lems. We study examples of both categories. In the next two sections, we describe
finite difference methods for these two types and then we discuss several other
methods for partial differential equations very briefly.


Initial value problems

An important example of this class of problems is the diffusion equation or the time-
dependent Schrödinger equation (seeSection 12.2.4for a discussion of the relation
between these two equations) which is mathematically the same as the diffusion
equation, the only difference being a factoribefore the time derivative. In the
following, we shall consider the Schrödinger equation, but the analysis is the same
for the diffusion equation.
Consider the one-dimensional time-dependent Schrödinger equation (we take
= 2 m=1):


i
∂ψ(x,t)
∂t

=−


∂^2 ψ(x,t)
∂x^2

+V(x)ψ(x,t), (A.64)

or, in a more compact notation:


i

∂ψ(x,t)
∂t

=Hψ(x,t) (A.65)

withHstanding for the Hamilton operator. First we discretiseHatLequidistant
positions on the real axis, with separationx. The value ofψat the pointxj=jx,
j=1,...,L, is denoted byψj. We can then approximate the second order derivative

Free download pdf