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 probl...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 0-method for 0 〈 θ ≤ 1, in both cases in maximum-norm, showing O（h2 ＋ 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 ＋ k3/2） error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.展开更多
文摘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 0-method for 0 〈 θ ≤ 1, in both cases in maximum-norm, showing O（h2 ＋ 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 ＋ k3/2） error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.
