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Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

Published online by Cambridge University Press:  03 June 2015

Bernard Bialecki  and
Ryan I. Fernandes
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: bbialeck@mines.edu
Email: rfernandes@pi.ac.ae
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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.

Keywords

65M70Alternating direction implicit methodorthogonal spline collocationtwo point boundary value problemCrank Nicolsonparabolic equationnon-rectangular region
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 461 - 476
Copyright
Copyright © Global-Science Press 2013

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References

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