Performance Analysis of Error-Control B-spline Gaussian Collocation Software for PDEs

Abstract

B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and partial differential equations (PDEs) in one space dimension (1D) for many years. The software package, BACOL, developed over a decade ago, was one of the first 1D PDE packages to provide both temporal and spatial error control. A new package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages implies that extensive testing and analysis of the test results is an essential factor in their development. In this paper, we investigate the performance of the BACOL and BACOLI packages with respect to several important machine independent algorithmic measures and examine the effectiveness of the new error estimation and error control strategies. We also investigate the influence of the choice of the degree of the B-splines on the efficiency and reliability of the solvers. These results will provide new insights into how to improve BACOLI, lead to improvements in the Gaussian collocation BVODE solvers, COLSYS and COLNEW, and guide the further development of B-spline Gaussian collocation software with error control for 2D PDEs.