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Volume 18, Issue 5
Convergence Analysis of ADI Orthogonal Spline Collocation Without Perturbation Terms

Bernard Bialecki & Ryan I. Fernandes

Int. J. Numer. Anal. Mod., 18 (2021), pp. 620-641.

Published online: 2021-08

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  • Abstract

For the heat equation on a rectangle and nonzero Dirichlet boundary conditions, we consider an ADI orthogonal spline collocation method without perturbation terms, to specify boundary values of intermediate solutions at half time levels on the vertical sides of the rectangle. We show that, at each time level, the method has optimal convergence rate in the $L^2$ norm in space. Numerical results for splines of orders 4, 5, 6 confirm our theoretical convergence rates and demonstrate suboptimal convergence rates in the $H^1$ norm. We also demonstrate numerically that the scheme without the perturbation terms is applicable to variable coefficient problems yielding the same convergence rates obtained for the heat equation.

  • Keywords

Convergence, alternating direction implicit method, orthogonal spline collocation, perturbation terms.

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-18-620, author = {Bialecki , Bernard and Fernandes , Ryan I.}, title = {Convergence Analysis of ADI Orthogonal Spline Collocation Without Perturbation Terms}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {5}, pages = {620--641}, abstract = {

For the heat equation on a rectangle and nonzero Dirichlet boundary conditions, we consider an ADI orthogonal spline collocation method without perturbation terms, to specify boundary values of intermediate solutions at half time levels on the vertical sides of the rectangle. We show that, at each time level, the method has optimal convergence rate in the $L^2$ norm in space. Numerical results for splines of orders 4, 5, 6 confirm our theoretical convergence rates and demonstrate suboptimal convergence rates in the $H^1$ norm. We also demonstrate numerically that the scheme without the perturbation terms is applicable to variable coefficient problems yielding the same convergence rates obtained for the heat equation.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/19385.html} }
TY - JOUR T1 - Convergence Analysis of ADI Orthogonal Spline Collocation Without Perturbation Terms AU - Bialecki , Bernard AU - Fernandes , Ryan I. JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 620 EP - 641 PY - 2021 DA - 2021/08 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/19385.html KW - Convergence, alternating direction implicit method, orthogonal spline collocation, perturbation terms. AB -

For the heat equation on a rectangle and nonzero Dirichlet boundary conditions, we consider an ADI orthogonal spline collocation method without perturbation terms, to specify boundary values of intermediate solutions at half time levels on the vertical sides of the rectangle. We show that, at each time level, the method has optimal convergence rate in the $L^2$ norm in space. Numerical results for splines of orders 4, 5, 6 confirm our theoretical convergence rates and demonstrate suboptimal convergence rates in the $H^1$ norm. We also demonstrate numerically that the scheme without the perturbation terms is applicable to variable coefficient problems yielding the same convergence rates obtained for the heat equation.

Bialecki , Bernard and Fernandes , Ryan I.. (2021). Convergence Analysis of ADI Orthogonal Spline Collocation Without Perturbation Terms. International Journal of Numerical Analysis and Modeling. 18 (5). 620-641. doi:
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