Given certain implicit difference approximations to u t = a ( x ) u x x + b ( x ) u x + c ( x ) u {u_t} = a(x){u_{xx}} + b(x){u_x} + c(x)u in the region − ∞ > x > ∞ , t ≧ 0 - \infty > x > \infty ,t \geqq 0 , with a finer x -mesh width in the left half-plane than in the right, we consider the stability in the maximum norm of these schemes using several different interface conditions (at x = 0 x = 0 ). In order to obtain our results, we first prove a stability theorem for certain simple second-order parabolic initial boundary systems on an evenly spaced mesh in the right half-plane alone. By a standard procedure, the first problem is converted into the second one, and solved in this manner.