Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T13:11:26.338Z Has data issue: false hasContentIssue false

Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

Published online by Cambridge University Press:  03 June 2015

Bernard Bialecki*
Affiliation:
Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401, USA
Ryan I. Fernandes*
Affiliation:
Department of Mathematics, The Petroleum Institute, Abu Dhabi, United Arab Emirates
*
Corresponding author. Email: [email protected]
Get access

Abstract

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles. When the ADI technique is coupled with orthogonal spline collocation (OSC) for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order. In [2], we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms. A natural question that arises is: Does this method have an extension to non-rectangular regions? In this paper, we present a simple idea of how the ADI OSC technique can be extended to some such regions. Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem (TPBVP). We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.

Type
Research Article
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bialecki, B. and Fernandes, R. I., An orthogonal spline collocation alternating direction implicit Crank-Nicolson method for linear parabolic problems on rectangles, SIAM J. Numer. Anal., 36 (1999), pp. 14141434.Google Scholar
[2]Bialecki, B. and Fernandes, R. I., An alternating-direction implicit orthogonal spline collocation scheme for nonlinear parabolic problems on rectangular polygons, SIAM J. Sci. Comput., 28 (2006), pp. 10541077.Google Scholar
[3]Buzbee, B. L., Dorr, F. W., George, J. A. and Golub, G. H., The direct solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal., 8 (1971), pp. 722736.CrossRefGoogle Scholar
[4]Diaz, 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. 358375.Google Scholar
[5]Diaz, J. C., Fairweather, G. and Keast, P., Algorithm 603 COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software, 9 (1983), pp. 376380.Google Scholar
[6]Douglas, J. Jr, and Dupont, T., Collocation methods for parabolic equations in a single space variable, Lecture Notes in Mathematics, 385 (1974), Springer-Verlag, New York.Google Scholar
[7]Fernandes, R.I., Bialecki, B. and Fairweather, G., Alternating direction implicit orthogonal spline collocation methods for evolution equations, Mathematical Modelling and Applications to Industrial Problems (MMIP-2011), Jacob, M.J. and Panda, S., eds., Macmillan Publishers India Ltd., 2012, pp. 311.Google Scholar
[8]Kincaid, D. and Cheney, W., Numerical Analysis, Brooks/Cole Publishing Company, California, 1991.Google Scholar
[9]Peaceman, D. W. and Rachford, H. H. Jr., The numerical solution of elliptic and parabolic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), pp. 2841.Google Scholar