Go to the ANGEL web site and complete Reading Assignment 1. Come prepared to discuss what you've read.
Most rigorous studies to quantify error associated with the selected mesh or time step sizes are based on Richardson Extrapolation. This started as a means of improving the accuracy of numerical solutions to differential equations, but also can be used as a basis for estimating errors associated with selection of the mesh and time step size. Without understanding these errors, speculation on the quality of various physical models associated with a reactor safety code is on shaky ground. If your mesh is reasonably fine, and you know the order of accuracy of your method, you can use Richardson Extrapolation and results from two different grids (or two different time steps) to say something about the error. If you don't know the methods accuracy, or don't have confidence that your mesh is fine enough, you need a study with at least three different spatial divisions (or time step sizes).
and time step sensitivity studies lead to an estimate of error
discretization, and are also important in procedures used to detect
errors. Roache and Oberkampf
have good discussions of this error analysis. It basically boils
fitting a curve to a sequence of results and extrapolating beyond those
to estimate the limiting answer with zero mesh length or time
Consider a sequence of three mesh lengths or time step sizes (from
largest) h1, h2,
and h3. Normally the sequence is generated with a
h3/ h2 = h2/h1.
Let f1, f2, and f3 be the computed results at the same point in space and time for the three corresponding values of h. Taking a clue from truncation error analysis, we look for an expression for f as a function of h in the form:
f(h) = fexact + a hp
f(h) = fexact + a hp
subtracting the equations in pairs gives
Given a value of p, equations for the two finest meshes can be solved for the remaining unknowns.
As a result the error on the finest mesh can be estimated as:
that if you have faith in the value of p obtained from a
Roache, P.J., “Verification
Validation in Computational Science and Engineering,” Hermosa
pp. 403-412, 1998.
Oberkampf, W. L. and Trucano, T. G., “Verification and validation in Computational Fluid Dynamics, Sandia National Laboratory Report SAND2002-059, 2002.