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BACOL: B-spline adaptive collocation software for 1-D parabolic PDEs

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Published:01 December 2004Publication History
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Abstract

BACOL is a new, high quality, robust software package in Fortran 77 for solving one-dimensional parabolic PDEs, which has been shown to be significantly more efficient than any other widely available software package of the same class (to our knowledge), especially for problems with solutions exhibiting rapid spatial variation. A novel feature of this package is that it employs high order, adaptive methods in both time and space, controlling and balancing both spatial and temporal error estimates. The software implements a spline collocation method at Gaussian points, with a B-spline basis, for the spatial discretization. The time integration is performed using a modification of the popular DAE solver, DASSL. Based on the computation of a second, higher order, global solution, a high quality a posteriori spatial error estimate is obtained after each successful time step. The spatial error is controlled by a sophisticated new mesh selection algorithm based on an equidistribution principle. In this article we describe the overall structure of the BACOL package, and in particular the modifications to the DASSL package that improve its performance within BACOL. An example is provided in the online Appendix to illustrate the use of the package.

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  1. BACOL: B-spline adaptive collocation software for 1-D parabolic PDEs

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          Mike Minkoff

          This paper presents the culmination of software development for one-dimensional parabolic partial differential equations (PDEs), which many people have contributed to over the past three decades. The described program is based on the use of high quality software, in a combination of different areas that are involved in developing a unified package for solving these one-dimensional problems. These areas include spline representation, ordinary differential equation (ODE) software integration, differential/algebraic methods, and mesh adaptation methods. This combined software package addresses the need for a package that simultaneously deals with spatial and temporal error control and the numerical solution of such problems. The paper, however, is about more than the software integration (which is well described in this and the companion online appendix). It provides a well-balanced history of development for the nonspecialist, with a derivation of error analysis for temporal and spatial approximation. In order to address issues of performance and efficiency, the authors have modified the differential algebraic system solver (DASSL) software, and exploit the linear algebraic structure of the almost block diagonal matrices involved. In addition to providing numerical examples, the paper has a clear explanation of the software structure and user-supplied routines. The paper concludes with a range of projects for future work. Online Computing Reviews Service

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