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.
Supplemental Material
Available for Download
This is the 1st online appendix for BACOL: B-spline adaptive collocation software for 1-D parabolic PDEs. The appendix supports the information on page 454.
- Adjerid, S., Flaherty, J. E., Moore, P. K., and Wang, Y. 1992. High-order adaptive methods for parabolic systems. Physica D 60, 94--111. Google Scholar
- Ascher, U., Christiansen, J., and Russell, R. D. 1981. Collocation software for boundary value ODEs. ACM Trans. Math. Softw. 7, 209--222. Google Scholar
- Berzins, M., Capon, P. J., and Jimack, P. K. 1998. On spatial adaptivity and interpolation when using the method of lines. Appl. Num. Math. 26, 117--133. Google Scholar
- Berzins, M. and Dew, P. M. 1991. Algorithm 690: Chebyshev polynomial software for elliptic-parabolic systems of PDEs. ACM Trans. Math. Softw. 17, 178--206. Google Scholar
- Berzins, M., Dew, P. M., and Furzeland, R. M. 1989. Developing software for time-dependent problems using the method of lines and differential-algebraic integrators. Appl. Num. Math. 5, 375--397. Google Scholar
- Blom, J. and Zegeling, P. 1994. Algorithm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations. ACM Trans. Math. Softw. 20, 194--214. Google Scholar
- Brenan, K. E., Campbell, S. L., and Petzold, L. R. 1996. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadephia.Google Scholar
- de Boor, C. 1977. Package for calculating with B-Splines. SIAM J. Numer. Anal. 14, 441--472.Google Scholar
- de Boor, C. 1978. A Practical Guide to Splines. Springer-Verlag, New York.Google Scholar
- Diaz, J. C., Fairweather, G., and Keast, P. 1983. FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination. ACM Trans. Math. Softw. 9, 358--375. Google Scholar
- Huang, W. and Russell, R. D. 1996. A moving collocation method for solving time dependent partial differential equations. Appl. Numer. Math. 20, 101--116. Google Scholar
- Keast, P. and Muir, P. H. 1991. Algorithm 688. EPDCOL: A more efficient PDECOL code. ACM Trans. Math. Softw. 17, 153--166. Google Scholar
- Leimkuhler, B., Petzold, L., and Gear, C. W. 1991. Approximation methods for the consistent initialization of differential-algebraic equations. SIAM J. Numer. Anal. 28, 205--226. Google Scholar
- Madsen, N. K. and Sincovec, R. F. 1979. Algorithm 540. PDECOL, general collocation software for partial differential equations. ACM Trans. Math. Softw. 5, 326--351. Google Scholar
- Moore, P. K. 1995. Comparison of adaptive methods for one dimensional parabolic systems. Appl. Numer. Math. 16, 471--488. Google Scholar
- Moore, P. K. 2001. Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension. Numer. Math. 90, 149--177.Google Scholar
- Nokonechny, T. B., Keast, P., and Muir, P. H. 1996. A method of lines package, based on monomial spline collocation, for systems of one dimensional parabolic differential equations. In Numerical Analysis (A.R. Mitchell 75th birthday volume). World Scientific Publishing, Singapore, 207--224.Google Scholar
- Petzold, L. R. 1986. A description of DASSL: A differential/algebraic system solver. Tech. rep., Sandia Labs, Livermore, CA.Google Scholar
- Petzold, L. R. and Lötstedt, P. 1986. Numerical solution of nonlinear differential equations with algebraic constraints II: Practical implications. SIAM J. Sci. Stat. Comput. 7, 720--733. Google Scholar
- Wang, R. 2002. High order adaptive collocation software for 1-D parabolic PDEs. Ph.D. thesis, Dalhousie University, http://www.mathstat.dal.ca/∼keast/research/grad_super.html. Google Scholar
- Wang, R., Keast, P., and Muir, P. 2004a. A comparison of adaptive software for 1-D parabolic PDEs. J. Comput. Appl. Math. 169, 127--150. Google Scholar
- Wang, R., Keast, P., and Muir, P. 2004b. A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. Appl. Numer. Math. 50, 239--260. Google Scholar
Index Terms
BACOL: B-spline adaptive collocation software for 1-D parabolic PDEs
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