Performance Analysis Results for Error Control 
            B-spline Gaussian Collocation PDE Solvers
  
             Jack Pew,  Connor Tannahill,   Paul Muir
  
ABSTRACT
B-spline Gaussian collocation software has been widely used for the numerical solution of 
PDEs in one space dimension (1D) for several decades.  Such packages represent the 
approximate solution as a linear combination  of B-spline basis functions, of a given 
degree p, with time-dependent coefficients which are determined by requiring the
approximate solution to satisfy the boundary  conditions and the differential equations 
at certain points within the interior of the spatial domain. An essential capability of a high 
quality numerical software package is that it provide error control. That is, the software 
must  return a numerical solution such that  an associated error estimate satisfies the 
given user tolerance.  The 1D PDE solver, BACOL, developed a little over a decade 
ago, was the first B-spline Gaussian collocation package to provide both temporal and 
spatial error control.  The recently developed package, BACOLI, improves upon the 
efficiency of the  spatial error estimation of BACOL through the use of two new 
interpolation-based  schemes and provides two corresponding types of spatial error 
control.  In this report we investigate the performances of the BACOL and BACOLI 
packages  with respect to a number of important algorithmic assessments and examine 
the  effectiveness of the new error estimation schemes, the new error control strategies, 
and the  influence of the choice of p on the efficiency of the solvers.  These results will 
provide insights for making improvements to BACOLI, for making improvements to 
Gaussian collocation software for boundary value ODEs, and for the further development 
of error control B-spline Gaussian collocation software  for the numerical solution of 2D 
PDEs.