An improvement of the algorithm suggested in the appendix of the author's and \textit{M. Roland's} paper [ibid. 17, No. 2, 35-47 (1993; Zbl 0786.65080)], is briefly outlined. The algorithm represents a numerical method for solving two-dimensional nonlinear advection-diffusion equations in a domain limited by a moving boundary. A theoretical analysis reveals that, in most cases, the velocity of the boundary must be predicted to be zero. This undesirable behaviour stems from the incompatibility of the nonlinear numerical stability condition and the criterion allowing the boundary to move. Therefore the improvement of the suggested algorithm is useful.