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Volume 5, Issue 4
Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions

Bernard Bialecki & Ryan I. Fernandes

Adv. Appl. Math. Mech., 5 (2013), pp. 461-476.

Published online: 2013-08

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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.

  • AMS Subject Headings

65M70

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COPYRIGHT: © Global Science Press

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@Article{AAMM-5-461, author = {Bialecki , Bernard and Fernandes , Ryan I.}, title = {Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {4}, pages = {461--476}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.13-13S05}, url = {http://global-sci.org/intro/article_detail/aamm/80.html} }
TY - JOUR T1 - Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions AU - Bialecki , Bernard AU - Fernandes , Ryan I. JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 461 EP - 476 PY - 2013 DA - 2013/08 SN - 5 DO - http://doi.org/10.4208/aamm.13-13S05 UR - https://global-sci.org/intro/article_detail/aamm/80.html KW - Alternating direction implicit method, orthogonal spline collocation, two point boundary value problem, Crank Nicolson, parabolic equation, non-rectangular region. AB -

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.

Bernard Bialecki & Ryan I. Fernandes. (1970). Alternating Direction Implicit Orthogonal Spline Collocation on Non-Rectangular Regions. Advances in Applied Mathematics and Mechanics. 5 (4). 461-476. doi:10.4208/aamm.13-13S05
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