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B-Spline Gaussian Collocation Software for Two-Dimensional Parabolic PDEs

Published online by Cambridge University Press:  03 June 2015

Zhi Li  and
Paul Muir
Zhi Li*
Affiliation:
Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, Canada B3H 3C3
Paul Muir*
Affiliation:
Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia, Canada B3H 3C3
*
Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

In this paper we describe new B-spline Gaussian collocation software for solving two-dimensional parabolic partial differential equations (PDEs) defined over a rectangular region. The numerical solution is represented as a bi-variate piecewise polynomial (using a tensor product B-spline basis) with time-dependent unknown coefficients. These coefficients are determined by imposing collocation conditions: the numerical solution is required to satisfy the PDE and boundary conditions at images of the Gauss points mapped onto certain subregions of the spatial domain. This leads to a large system of time-dependent differential algebraic equations (DAEs) which is solved using the DAE solver, DASPK. We provide numerical results in which we use the new software, called BACOL2D, to solve three test problems.

Keywords

65M2065M70CollocationB-splinestwo-dimensional partial differential equationsdifferential-algebraic equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 528 - 547
Copyright
Copyright © Global-Science Press 2013

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References

[1]Arsenault, T., Smith, T. and Muir, P.H., Superconvergent interpolants for efficient spatial error estimation in 1D PDE collocation solvers, Can. Appl. Math. Q., 17 (2009), pp. 409–431.Google Scholar
[2]Arsenault, T., Smith, T., Muir, P. H. and Pew, J., Asymptotically exact interpolation-based error estimates for collocation solutions of 1D PDEs, to appear, Can. Appl. Math. Q., 2012.Google Scholar
[3]Boor, C. De, A practical guide to splines, Applied Mathematical Sciences, 27, Springer-Verlag, New York, 1978.Google Scholar
[4]Brenan, K. E., Campbell, S. L. and Petzold, L. R., Numerical solution of initial-value problems in differential-algebraic equations, Society for Industrial and Applied Mathematics, Philadelphia, 1989.Google Scholar
[5]Brown, P. N., Hindmarsh, A. C. and Petzold, L. R., Using Krylov methods in the solution of large-scale differential-algebraic systems, SIAM J. Sci. Comput., 15 (1994), pp. 1467–1488.CrossRefGoogle Scholar
[6]Cao, W., Huang, W. and Russell, R. D., A study of monitor functions for two-dimensional adaptive mesh generation, SIAM J. Sci. Comput., 20 (1999), pp. 1978–1994.Google Scholar
[7]Cerutti, J. H. and Parter, S. V., Collocation methods for parabolic partial differential equations in one space dimension, Numer. Math., 26 (1976), pp. 227–254.CrossRefGoogle Scholar
[8]Christara, C. C., Quadratic spline collocation methods for elliptic partial differential equations, BIT, 34 (1994), pp. 33–61.Google Scholar
[9]Dembo, R. S., Eisenstat, S. C. and Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), pp. 400–408.Google Scholar
[10]Díaz, J.C., Fairweather, G. and Keast, P., FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software, 9 (1983), pp. 358–375.Google Scholar
[11]Gockenbach, M. S., Partial differential equations: analytical and numerical methods, Society for Industrial and Applied Mathematics, Philadelphia, 2002.Google Scholar
[12]Houstis, E. N., Vavalis, E. A. and Rice, J. R., Convergence of O(h 4) cubic spline collocation methods for elliptic partial differential equations, SIAM J. Numer. Anal., 25 (1988), pp. 54–74.CrossRefGoogle Scholar
[13]Houstis, E. N., Mitchell, W. F. and Rice, J. R., Algorithm 637: GENCOL: collocation of general domains with bicubic hermite polynomials, ACM Trans. Math. Software, 11 (1985), pp. 413–415.Google Scholar
[14]Houstis, E. N., Mitchell, W. F. and Rice, J. R., Algorithm 638: INTCOL and HERMCOL: collocation on rectangular domains with bicubic hermite polynomials, ACM Trans. Math. Software, 11 (1985), pp. 416–418.Google Scholar
[15]Huang, W. and Sloan, D. M., A simple adaptive grid method in two dimensions, SIAM J. Sci. Comput., 15 (1994), pp. 776–797.Google Scholar
[16]Huang, W., Practical aspects of formulation and solution of moving mesh partial differential equations, J. Comput. Phys., 171 (2001), pp. 753–775.Google Scholar
[17]Huang, W. and Russell, R. D., Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comput., 20 (1998), pp. 998–1015.Google Scholar
[18]Huang, W. and Russell, R. D., Adaptive moving mesh methods, Applied Mathematical Sciences, 174, Springer, New York, 2011.Google Scholar
[19]Hundsdorfer, W. and Verwer, J., Numerical solution of time-dependent advection-diffusion-reaction equations, Springer Series in Computational Mathematics, 33, Springer-Verlag, Berlin, 2003.Google Scholar
[20]Leveque, R. J., Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 2007.Google Scholar
[21]Li, Z., B-spline Collocation Methods for Two-Dimensional, Time-Dependent PDEs, M.Sc. thesis, Saint Mary’s University, 2012.Google Scholar
[22]Mattheij, R. M. M., Rienstra, S. W. and Boonkkamp, J. H. M. T. T., Partial differential equations: modeling, analysis, computation, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, 2005.Google Scholar
[23]Ng, K. S., Spline Collocation on Adaptive Grids and Non-Rectangular Regions, Ph.D. thesis, University of Toronto, 2005.Google Scholar
[24]Pani, A. K., Fairweather, G. and Fernandes, R. I., ADI orthogonal spline collocation methods for parabolic partial integro-differential equations, IMA J. Numer. Anal., 30 (2010), pp. 248–276.Google Scholar
[25]Russell, R. D. and Sun, W., Spline collocation differentiation matrices, SIAM J. Numer. Anal., 34 (1997), pp. 2274–2287.Google Scholar
[26]Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461–469.Google Scholar
[27]Saad, Y., Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics, Philadelphia, 2003.Google Scholar
[28]Sun, W., B-spline collocation methods for elasticity problems, in Scientific Computing and Applications, Adv. Comput. Theory Pract., 7 (2000), pp. 133–141.Google Scholar
[29]Velivelli, A. C. and Bryden, K. M., Parallel performance and accuracy of lattice Boltzmann and traditional finite difference methods for solving the unsteady two-dimensional Burger’s equation, Physica A: Statistical Mechanics and its Applications, 362 (2006), pp. 139–145.Google Scholar
[30]Wang, R., Keast, P. and Muir, P. H., BACOL: B-spline adaptive collocation software for 1-D parabolic PDEs, ACM Trans. Math. Software, 30 (2004), pp. 454–470.CrossRefGoogle Scholar
[31]Wang, R., Keast, P. and Muir, P. H., A high-order global spatially adaptive collocation method for 1-D parabolic PDEs, Appl. Numer. Math., 50 (2004), pp. 239–260.Google Scholar
[32]Wang, R., Keast, P. and Muir, P. H., A comparison of adaptive software for 1D parabolic PDEs, J. Comput. Appl. Math., 169 (2004), pp. 127–150.Google Scholar
[33]Wang, Y., A Parallel Collocation Method for Two Dimensional Linear Parabolic Separable Partial Differential Equations, Ph.D. thesis, Dalhousie University, 1995.Google Scholar
[34]Wendel, S., Maisch, H., Karl, H. and Lehner, G., Two-dimensional B-spline finite elements and their application to the computation of solitons, Electrical Engineering (Archiv fur Elektrotechnik), 76 (1993), pp. 427–435.Google Scholar