An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms

An alternating direction implicit (ADI) scheme is introduced which is capable of solving a general parabolic equation in two space dimensions with mixed derivative and convective terms. In the case of constant coefficients the scheme is shown to be unconditionally stable. The study was motivated by the investigation of flow in a dye laser cell (a device used for the amplification of a laser beam), a simple model for which involves laminar flow in a two-dimensional symmetric channel subject to impulsive heating. Numerical results are presented for this problem, and the qualitative behaviour of the temperature distribution within the channel for different Peclet numbers is discussed.