An alternating direction implicit (A.D.I.) method, which requires the solution of two tridiagonal sets of equations at each time step, is derived for solving a parabolic equation with variable coefficients in two space dimensions with a mixed derivative. The method is shown to be unconditionally stable for two semi-infinite ranges of an auxiliary parameter. Other existing finite difference schemes are mentioned and numerical results are presented.
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S. McKee (1970) studied this question.