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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]

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems. [Copyright &y& Elsevier]

A nonoverlapping domain decomposition approach with uniform and matching grids is used to define and compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on a square partitioned into four squares. The collocation solution on four interfaces is computed using the preconditioned conjugate gradient method with the preconditioner defined in terms of interface preconditioners for the adjacent squares. The collocation solution on four squares is computed by a matrix decomposition method that uses fast Fourier transforms. With the number of preconditioned conjugate gradient iterations proportional to log2 N, the total cost of the algorithm is O( N2 log2 N), where the number of unknowns in the collocation solution is O( N2). The approach presented in this paper, along with that in [B.Bialecki and M.Dryja, A nonoverlapping domain decomposition method for orthogonal spline collocation problems. SIAM J. Numer. Anal. (2003) 41, 1709 – 1728.], generalizes to variable coefficient equations on rectangular polygons partitioned into many subrectangles and is well suited for parallel computation. [ABSTRACT FROM AUTHOR]