Read the material linked on the NucE
470 ANGEL page under "Reading Assignment 1",
and start HW5
The material linked from this page, and the contents of the
associated lecture provide a beginning to understanding the errors
associated with the use of discrete approximations to differential
equations. This will give you a concrete understanding of what
people mean by "truncation error" and "order of accuracy" for a
discrete approximation. Although the mathematics can get ugly,
the steps in this process are fairly simple to remember.
Basic analysis of the error associated with a finite volume or
finite difference method is relatively easy if you understand a Taylor
series expansion, and have a tool like Mathematica, Maple, or Mathcad
to do the tedious work for you. Start by graphing all of the
discrete points in space and time at which you will use (or generate)
information. For an explicit method it might look something like:
The most central of the points used is located at time level "n" and
volume number "i", so we declare the equation to be evaluated there,
look at everything else in terms of values obtained at that point.
This is done via Taylor series expansions about that point.
One example of this type analysis is in
some work by Tony Hirt, which provides further insight into the
stability of first order upwind methods. However, the primary
reason to remember Hirt's analysis is not simply error, but as a
quantitative way to compare the numerical diffusion in a code like
TRACE to the actual turbulent diffusion processes present in a
system. What you will find is that for mesh sizes and time steps
standardly used in reactor safety analysis, numerical diffusion is
significantly larger than turbulent diffusion. Another example of
error analysis can be
found in the section on "Truncation Error" in this conduction
n+1 | *
n | * * *
i-1 i i+1
I have provided some sample Mathematica
commands to do a simple error analysis on a semi-implicit mass
equation. If you want to use this as a method of error analysis,
open the file under Mathematica and type the Enter key
with the Shift held down to evaluate the commands. In the final
set of results you will see three terms that can be interpreted as the
non-conservative differential form of the mass equation.
Everything else is error. Note that the lowest powers of
time step size and mesh length define the order of accuracy of the
method. In this case the method is first order accurate in space
So What ?
Order of accuracy tells you how much improvement to expect if you
are close to a "good" answer and want to make it better. If my
method is first order accurate in space, and I double the number
of volumes in my system by cutting every existing volume in half, I can
expect my mesh induced error to drop by a factor of two. If my
method is second order accurate in space, and I split all volumes, I
can expect the mesh error to drop by a factor of 4. Similar
arguments hold for accuracy in time. Be aware that the effective order
of accuracy of a method determined from Richardson
extrapolation incorporates effects of higher order terms in the
Taylor series if the mesh is too large, and effects from boundary
conditions. TRACE for example has a second order difference
approximation to the conduction equation, but incorporates a first
order approximation to the convective boundary conditions. This
generally results in an effective order of accuracy that is less than 2.
You need to be aware that higher order accuracy is not without
drawbacks. If you are propagating a wave like we did in the second exercise, a basic higher
order method will tend to undershoot the answer just before the correct
wave front position and overshoot just behind the wave. This link is a simple density wave,
with a time step equal to a tenth of the material Courant limit.
Methods piloted in addition to first order upwind are either 2nd
(Leith, QUICK) or 3rd (QUICKEST) order accurate in space.
Fortunately, the problem can be mitigated. If you learn
that you're running a code with a higher order method (order > 1),
and you expect waves of some sort in your calculation (slugs of liquid
qualify), then make sure that the code has a "flux limiter" available
and engaged. Generally flux limiters will reduce the local
spatial accuracy to first order.
Remember that codes advertising higher order methods,
don't always live up to the claims.
Maintained by John Mahaffy : email@example.com