Stability condition for difference schemes for parabolic systems

Recently, Goldberg studied in [M. Goldberg, SIAM J. Numer. Anal., 35 (1998), pp. 478-493, M. Goldberg, SIAM J. Numer. Anal., 35 (1998), pp. 1995-2003] stability conditions of a well-known family of difference schemes for the following multidimensional parabolic system of second order: [GRAPHICS] Here we present a modified stability condition for the above problem, and prove that this condition is optimal in some sense.