Volume 5, Issue 4
The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

B. Bialecki, G. Fairweather & J. C. López-Marcos

Adv. Appl. Math. Mech., 5 (2013), pp. 442-460.

Published online: 2013-08

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  • Abstract

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

  • Keywords

Heat equation nonlocal boundary conditions orthogonal spline collocation Hermite cubic splines convergence analysis superconvergence

  • AMS Subject Headings

65N35 65N12 65N15

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COPYRIGHT: © Global Science Press

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@Article{AAMM-5-442, author = {B. Bialecki, G. Fairweather and J. C. López-Marcos}, title = {The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {4}, pages = {442--460}, abstract = {

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.13-13S03}, url = {http://global-sci.org/intro/article_detail/aamm/79.html} }
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