A Numerical Study of Global Error Estimation
Schemes for Defect Control BVODE Codes
Jason J. Boisvert, Paul H. Muir and Raymond J. Spiteri
A defect control boundary value ordinary differential equation (BVODE)
code computes a numerical solution for which an estimate of the defect,
i.e., the amount by which the approximate solution fails to satisfy the
BVODE, is less than a user-provided tolerance. Defect control is attractive
from a backward error viewpoint and the computation of the defect is
inexpensive. A number of defect control codes are now in wide use, including
the MATLAB codes, bvp4c and bvp5c, and the Fortran codes, MIRKDC
and BVP SOLVER. However, the global error of a numerical solution, i.e.,
the difference between the numerical solution and the exact solution, is
often more familiar to users, and it can therefore be useful for a defect
control code to also return an estimate of the global error for the defect
controlled numerical solution it computes. The ratio of the global error to
the maximum defect can also provide an indication of the conditioning of
the BVODE. BVP SOLVER currently provides an option for the return of a
global error estimate based on Richardson extrapolation, and in this paper
we consider the practical implementation within BVP SOLVER of three
alternative strategies based on the direct use of a higher order discretization
formula, an estimate of the BVODE conditioning constant, and a
deferred correction approach. We provide numerical results comparing
the four estimators that show that the approaches based on the direct
use of a higher order discretization formula and deferred correction are
superior to the other two approaches.