Performance Analysis of Interpolation-based Spatial Error Control B-spline Gaussian
Collocation PDE Software: BDF Time Integration vs. IRK Time Integration

Jack Pew, Tom Murtha, Connor Tannahill, Paul Muir

Abstract

This report considers BACOLI and BACOLRI, two recently developed
members of a family of software packages for the error controlled numerical 
solution of systems of one-dimensional partial differential equations
(PDEs). The two original members of this family, BACOL and BACOLR,
employ a spatial discretization scheme based on B-spline Gaussian collocation 
which is coupled with spatial error estimation and adaptive mesh
refinement to provide spatial error control. For the error controlled time
integration, BACOL employs a package called DASSL, which is based
on a family of Backward Differentiation Formulas (BDFs). BACOLR, a
modification of BACOL, replaces DASSL with RADAU5, which is based
on a fifth order implicit Runge-Kutta method. The first of the new packages, 
BACOLI, was developed from BACOL; the second new package, BACOLRI, was 
developed from BACOLR. The fundamental difference between each new package and 
their corresponding earlier versions is a more efficient spatial error estimation 
computation based on the introduction of two new interpolation-based schemes. In 
this report, we provide extensive numerical results to compare the performances 
of the two new packages. We show that BACOLRI has comparable performance to that 
of BACOLI on a standard set of test problems and that, for problems where BACOLI 
fails due to stability issues associated with the BDF time integration methods, 
it is able to efficiently compute error controlled numerical solutions.