University of the Witwatersrand
Advanced Techniques in Physics : Part 2 - 2004 Tutorial 2 : 55 Marks
In the context of discretising partial differential equations :
Give the difference formula for the Laplacian.
Using the Taylor Expansion, evaluate the error of
the approximation explicitly to the next order.
Suppose and are two solutions
with the mesh size and respectively.
Derive the Richardson extrapolation formula which
shows that an improved solution is
In the context of investigating the stability of numerical solutions :
Write down the explicit discretised version of the Schrödinger equation
directly from the differential equation.
Use Von Neumann stability analysis (based on the local solution
) to show
Show that and that this approach is unstable.
Show that the the explicit discretised version of the
Schrödinger equation corresponds to finite difference
approximation to the formal solution which is first order in time,
Show that the operator is not unitary.
An implicit scheme resulting from the discretised
version of the Schrödinger equation (after some re-arrangement)
is by contrast
which is also first order accurate in time and not unitary. Use
Von Neumann stability analysis again to show that
and that this is now unconditionally stable.
Cayley's approximation for the finite difference of
leads to a finite difference scheme which is stable, unitary and second
order accurate in both space and time.
Suggest why it would become second order accurate.
In the context of curve fitting :
Suppose both the dependent variables in a two dimensional curve had errors
Derive a first order approximation to the variance
of the function .
In the context of Monte Carlo Methods :
A sequence of random events has a constant average rate ,
an exponential waiting time distribution
for the time between each event and a Poisson frequency distribution for the
number of events in time .
links the two distributions, as is the average of the
Poisson frequency distribution for the number of events in time and
characterises the mean life time of the exponential waiting
This can be exploited to derive a sampling method for Poisson Deviates
and then setting where the are exponential deviates.