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 - <p style="text-align: justify;">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 $0 &lt; θ ≤ 1$, 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.</p>