Numerical solution of the heat equation with nonlocal boundary conditions

The θ-method for the heat equation with nonlocal boundary conditions is discussed in this paper. The unconditional stability is proved for θ⩾12, subject to a condition that is much weaker than the one assumed in a paper by Ekolin. Due to the nonlocal boundary conditions, the systems of linear equations generated by the θ-method have a coefficient matrix that is tridiagonal except its first and last rows. Three efficient algorithms for solving this kind of linear systems are presented. A simple numerical example is given to compare their efficiency.