Title:   Efficient Interpolation-based Error Estimation for 
              1D Time-Dependent PDE Collocation Codes

Authors: Tom Arsenault, Tristan Smith, Paul Muir, and Pat Keast

Abstract: This report describes some recent work on interpolation based
approaches to spatial error estimation when Gaussian collocation is 
employed as the spatial discretization method in a method-of-lines 
algorithm for the numerical solution of a system of one-dimensional 
parabolic partial differential equations (PDEs).  At certain points within
the problem domain, the collocation solution is superconvergent and this 
report describes how an interpolant based on these superconvergent
values can be used to provide an efficient error estimate for the collocation
solution. We also consider a second approach that involves an interpolant for
which the interpolation error is asymptotically equivalent to the error of 
the collocation solution. We implement these new schemes within a modified 
version of a parabolic PDE collocation solver, BACOL, a recently developed 
software package for the numerical solution of systems of 1D time-dependent 
parabolic PDEs. BACOL employs a high order spatial discretization scheme 
based on B-spline collocation. BACOL generates the spatial error estimate 
by computing two global collocation solutions to the PDEs, one based on 
B-splines of degree p and the other on B-splines of degree p+1.  The 
difference between the two collocation  solutions gives a high order 
estimate of the spatial error of the lower order collocation solution. The 
computation of the two global collocation solutions is obviously a 
significant computational expense; the two interpolation based approaches 
mentioned above provide low cost alternatives upon which to base the error
estimation scheme. Numerical results are provided to compare these new error
estimates with the one currently employed within BACOL.