A Computational Study of the Efficiency of Collocation Software for 
1D Parabolic PDEs with Interpolation-based Spatial Error Estimation 

Jack Pew, Zhi Li, Paul Muir

Abstract
BACOL is a software package for the numerical solution of systems
of one-dimensional parabolic partial differential equations (PDEs) that
has been shown to be superior to other similar packages, especially for
problems exhibiting sharp spatial layer regions where a stringent toler-
ance is imposed. BACOL, based on a method-of-lines algorithm, features
adaptive control of a high order estimate of the spatial error. (Adap-
tive control of the temporal error in the numerical solution of the system
of differential-algebraic equations (DAEs), arising from a B-spline Gaus-
sian collocation spatial discretization, is provided by the underlying DAE
solver, DASSL.) The spatial error estimate for the collocation solution
computed by the code is obtained by computing a second collocation so-
lution, which involves a substantial cost - the execution time and memory
usage are almost doubled.

In this report we discuss BACOLI, a new version of BACOL that computes 
only one collocation solution and uses efficient interpolation-based
approaches to obtain a spatial error estimate. These approaches have re-
cently been shown to provide spatial error estimates of comparable quality
to those computed by BACOL. We describe the substantial modification of 
the BACOL code that was required in order to obtain BACOLI and provide 
numerical results to compare BACOL and BACOLI. We show that BACOLI is 
about twice as efficient as BACOL. 

Subject Classification: 65M15, 65M20, 65M70
Keywords: Efficiency, 1D Parabolic PDEs, Collocation, Spatial Error Estima-
tion, Interpolation, Method-of-Lines.

This work was supported by the Mathematics of Information Technology and Complex
Systems Network, the Natural Sciences and Engineering Research Council of Canada and
Saint Mary’s University.

Jack Pew: Saint Mary’s University, Halifax, NS, Canada, B3H 3C3
ZHi Li: Hong Kong Baptist University, Kowloon Tong, Kowloon
Paul Muir: Saint Mary’s University, Halifax, NS, Canada, B3H 3C3