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

V. Thomée & A.S. Vasudeva Murthy

DOI: 10.4208/jcm.1406-m4443

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 $O(h^2$ + $k^{3/2}$) 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 $O(h^2$ + $k^{3/2}$) 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, Artificial boundary conditions, unbounded domains, 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 $O(h^2$ + $k^{3/2}$) error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.

Thomée , V. and Vasudeva Murthy , A.S.. (2015). 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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