Volume 33, Issue 1
Finite Difference Methods for the Heat Equation with a Nonlocal Boundary Condition

V. Thomée & A.S. Vasudeva Murthy

J. Comp. Math., 33 (2015), pp. 17-32.

Published online: 2015-02

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  • Abstract

We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the θ-method for 0 < θ ≤ 1, in both cases in maximum-norm, showing O(h^2 + k) error bounds, where h is the mesh-width and k the time step. We then give an alternative analysis for the case θ = 1/2, the Crank-Nicolson method, using energy arguments, yielding a O(h² + k^{3/2}) error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.

  • Keywords

Heat equation Artificial boundary conditions unbounded domains product quadrature

  • AMS Subject Headings

65M06 65M12 65M15.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

thomee@chalmers.se (V. Thomée)

vasu@math.tifrbng.res.in (A.S. Vasudeva Murthy)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-17, author = {Thomée , V. and Vasudeva Murthy , A.S. }, title = {Finite Difference Methods for the Heat Equation with a Nonlocal Boundary Condition}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {1}, pages = {17--32}, abstract = {

We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the θ-method for 0 < θ ≤ 1, in both cases in maximum-norm, showing O(h^2 + k) error bounds, where h is the mesh-width and k the time step. We then give an alternative analysis for the case θ = 1/2, the Crank-Nicolson method, using energy arguments, yielding a O(h² + k^{3/2}) error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1406-m4443}, url = {http://global-sci.org/intro/article_detail/jcm/9825.html} }
TY - JOUR T1 - Finite Difference Methods for the Heat Equation with a Nonlocal Boundary Condition AU - Thomée , V. AU - Vasudeva Murthy , A.S. JO - Journal of Computational Mathematics VL - 1 SP - 17 EP - 32 PY - 2015 DA - 2015/02 SN - 33 DO - http://doi.org/10.4208/jcm.1406-m4443 UR - https://global-sci.org/intro/article_detail/jcm/9825.html KW - Heat equation KW - Artificial boundary conditions KW - unbounded domains KW - product quadrature AB -

We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the θ-method for 0 < θ ≤ 1, in both cases in maximum-norm, showing O(h^2 + k) error bounds, where h is the mesh-width and k the time step. We then give an alternative analysis for the case θ = 1/2, the Crank-Nicolson method, using energy arguments, yielding a O(h² + k^{3/2}) error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.

V. Thomée & A.S. Vasudeva Murthy . (2020). Finite Difference Methods for the Heat Equation with a Nonlocal Boundary Condition. Journal of Computational Mathematics. 33 (1). 17-32. doi:10.4208/jcm.1406-m4443
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