Stability in the numerical solution of the heat equation with nonlocal boundary conditions Original Research Article

This paper deals with numerical methods for the solution of the heat equation with integral boundary conditions. Finite differences are used for the discretization in space. The matrices specifying the resulting semidiscrete problem are proved to satisfy a sectorial resolvent condition, uniformly with respect to the discretization parameter. Using this resolvent condition, unconditional stability is proved for the fully discrete numerical process generated by applying A(θ)-stable one-step methods to the semidiscrete problem. This stability result is established in the maximum norm; it improves some previous results in the literature in that it is not subject to various unnatural restrictions which were imposed on the boundary conditions and on the one-step methods.