An Alternating Direction Implicit Backward Differentiation Orthogonal Spline Collocation Method for Linear Variable Coefficient Parabolic Equations

We present a new fully discrete method to solve a linear variable coefficient parabolic equation on a rectangular region. The scheme is an alternating direction implicit method which uses the third order backward differentiation formula for the time discretization and piecewise Hermite bicubic orthogonal spline collocation for the spatial discretization. The $L^2$ norm stability and convergence analysis is carried out for the heat equation. The stability analysis reveals that the scheme is stable with respect to the right-hand sides but it is unstable with respect to the initial conditions. We explain, using a damping property of the scheme and round-off error analysis, why this instability is harmless. We also show how a different choice of initial values leads to optimal convergence orders. Numerical results demonstrate the third order accuracy in time and optimal order accuracy in space using maximum and Sobolev norms, respectively.

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