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For the heat equation on a rectangle, we consider the finite difference ADI method without a perturbation term on vertical sides for the intermediate solution. Using stability results of Andreev [1, 2] for the discrete harmonic function we prove, except for a factor, the second order bound that is stated without a proof by Samarski [8]. [ABSTRACT FROM AUTHOR]

Abstract: Since the invention by Peaceman and Rachford, more than 60 years ago, of the well celebrated ADI finite difference scheme for parabolic initial-boundary problems on rectangular regions, many papers have been concerned with prescribing the boundary values for the intermediate approximations at half time levels in the case of nonzero Dirichlet boundary conditions. In the present paper, for variable coefficient parabolic problems and time-stepsize sufficiently small, we prove second order accuracy in the discrete norm of the ADI finite difference scheme in which the intermediate approximations do not involve the so called "perturbation term". As a byproduct of our stability analysis we also show that, for variable coefficients and time-stepsize sufficiently small, the ADI scheme with the perturbation term converges with order two in the discrete norm. Our convergence results generalize previous results obtained for the heat equation. [ABSTRACT FROM AUTHOR]